Download Non-relativistic Holography and Renormalization

Non-relativistic Holography and
Renormalization
Non-relativistic Holography and
RenormalizationSupervisor: Prof. dr. Jan de Boer
Author: Dimitrios Korres
Master’s Thesis
Author: Dimitrios Korres
MSc in Physics
Track:
Supervisor: Prof. Jan de Boer
Theoretical
MSc Thesis Physics
Faculteit
der Natuurwetenschappen, Wiskunde
en Informatica
Faculteit
der Natuurwetenschappen,
Wiskunde
en Informatica
Institute
for Theoretical
Physics
Instituut
voor
Theoretische
Fysica
Universiteit van Amsterdam
Universiteit
van Amsterdam
Valckenierstraat 65
1018 XE Amsterdam, the Netherlands
i
Abstract
The present thesis reviews the recently proposed non-relativistic
holographic duality and the application of the holographic renormalization procedure to this case is discussed. After a brief description of the renormalization group, we comment on the Hamilton
constraints in General Relativity. We then explain the AdS/CFT
correspondence and combine everything up to that point to describe the holographic renormalization procedure. Afterwards, we
discuss aspects of non-relativistic holography which was put forth
by Son and independently by Balasubramanian and McGreevy.
We finally attempt to bring the holographic renormalization group
technology in the non-relativistic framework. Some developments
are described and we make some suggestions on how this program
could succeed.
Contents
Contents
iii
Historical Overview – Motivation
1
1 Renormalization
1.1 Perturbative Renormalization - A Toy
1.2 Wilsonian Renormalization Theory .
1.3 Renormalization Group Flows
. .
1.4 The Callan-Symanzik Equation . .
1.5 Renormalization Schemes . . . .
Scheme (in)dependence
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Model
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2 Action Principle and Geometry
2.1 Hamilton-Jacobi Theory . . . . .
2.2 Electromagnetism and the Hamiltonian
2.3 The ADM Formalism . . . . . .
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3 AdS/CFT Correspondence
3.1 A first hint
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Conformal Invariance .
Anti-de Sitter Space
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3.2 String Theory
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3.3 The AdS/CFT Duality .
3.4 The AdS/CFT Dictionary
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4 Holographic Renormalization
4.1 Hamilton-Jacobi applied to gravity . .
4.2 The Holographic Renormalization Group
Callan-Symanzik equation . . . . .
Gauge Fields and Hamilton-Jacobi . .
5 Non-relativistic Holography
5.1 Schrödinger Group . . .
5.2 Holographic Construction .
Causal Structure and Global
5.3 The Bulk Action
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5.4 String Theory Embedding .
Null Melvin Twist . . .
5.5 The 5d effective Lagrangian
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Coordinates
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iv
Contents
6 Hamilton-Jacobi and non-relativistic holography
6.1 Non-relativistic case
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6.2 Application . . . . . . . . . . . .
6.3 Domain wall solutions . . . . . . . . .
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7 Discussion
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Acknowledgements
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Appendices
85
A Dimensional Regularization
85
B Anti-de Sitter space
87
C Null Melvin Twist
C.1 Complex Projective Space . .
C.2 Hopf Fibration . . . . .
C.3 Buscher Rules and Conventions
C.4 Algorithm Steps . . . . .
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D Kaluza-Klein Compactifications
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Bibliography
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Historical Overview –
Motivation
Mathematics and Physics have been complementary to each other for the past
few centuries. Intuitions and tools developed in one field often turned out to be
groundbreaking and extremely useful in the other. In the previous century, the
two areas of knowledge became inextricably linked. The astonishing progress
in both fields, triggered by amazing discoveries during the first part of the
century, changed our view of the world in a profound and unprecedented way.
Perhaps it is no accident that the two fields seem to be following similar paths.
Mathematician David Hilbert lived in that exciting era of great scientific
discoveries, which initiated a scientific revolution based on radical new ideas.
He was the ambassador of the notion and belief that mathematics should be
a complete theory–a system of knowledge complete in itself. In his famous
lecture of 1900 in Paris, he posed twenty three unsolved problems that he
believed were important in determining the logical foundations of science.
However, his vision of developing a system with which one could attack any
problem and answer any question, was short-lived. In 1931, a young mathematician named Kurt Gödel, published a paper that shuttered the hope for
a complete structure that would be entirely self-contained and self-consistent.
In his incompleteness theorems he showed that it is impossible to constructively show that an axiomatic theory is consistent, and also that in a consistent
theory, there will be theorems that cannot be proved. He showed that it was
as if mathematicians wanted to settle something that had the same nature as
Epiminides paradox, who said that “all Cretans are liars”. Being from Crete
himself, this puts forth an obvious inconsistency of that statement.
Interestingly, four years earlier, physicist Werner Heisenberg had made a
bold statement that goes by the name of Uncertainty Principle ∗ . So, in a
similar fashion, physicists had already started getting used to the idea that
we cannot know everything; not simultaneously at least. In both fundamental
mathematics and physics, what we can or cannot know became a deep and
important question.
Albert Einstein on the other hand, remained until the end of his life, a
strong advocate of the need for a complete, unified theory of nature. To the
present day, this is still a goal for a large number of theoretical physicists.
From the modern point of view, the Standard Model of fundamental particles
provides a satisfactory and complete understanding of the world–despite some
problems–only if we exclude gravity. It is the pinnacle of modern science
and was established in the early 1970’s. The remaining piece of the puzzle
∗
Which is not really a principle since it is a statement that can be proved.
1
2
Historical Overview – Motivation
however, seems to be a quantum field theory of gravity that can somehow
be incorporated in the Standard Model. In spite of many years of intense
research, this remains unattainable. However, there have been some important
developments, mainly in the context of string theory in the past decades.
Meanwhile, in the mathematics front, a new paradigm was emerging. New
hope appeared when in the 1960’s Robert Langlands posed a new program for
a unification of seemingly different and disjoint areas of mathematics. Inspired
by the Taniyama-Shimura conjecture he suggested what is now known as the
“Langlands program”. The Taniyama-Shimura conjecture gained its recent
fame because of its role in the proof of “Fermat’s last theorem” in the previous
decade. Nonetheless, its implications and influence in mathematics has been
much greater than that. The idea behind the Langlands program is that
separate areas of mathematics can be dual (equivalent) to each other. This
notion was something that physicists first came across in Maxwell’s theory of
electromagnetism.
Surprisingly, theoretical physicists came to a similar situation once again.
Until 1995 string theory appeared to be not one, but five different unified
theories of nature! How could all five of them be correct? We only observe
one universe, and it should be described by only one theory. This was an
embarrassing problem in string theory. Following the legacy of the Langlands
program, Edward Witten pushed an existing intricate web of dualities further
and showed that all five, seemingly different, string theories were different
limits of the one and same theory; it was dubbed M-theory and this discovery
is known as the “second superstring revolution”.
The rekindled interest in dualities and the accumulation of knowledge and
experience led, two years later, to an important discovery. In the seminal paper
by Maldacena, it was conjectured that string theory, which includes gravity,
is equivalent to a quantum field theory without gravity on the boundary of
this space. Because the boundary has one or less dimensions, it also became
linked to the holographic principle proposed a few years earlier by Gerard
’t Hooft and independently by Leonard Susskind. The idea of holography
dates back to Plato’s cave, but the realization of a rigorous, scientific theory
with predictive aspirations remains one of the most important and inspiring
developments in recent years. Perhaps, this does not embody the hopes for
a novel, exotic idea for a quantum field theory of gravity that some had. If
the conjecture is proved to be correct, it essentially gives an interpretation
of a quantum field theory of gravity with terms and notions that are already
known and studied–what we could call “traditional” field theory. In this sense,
the conjecture seems to render the aim for a completely new and extravagant
idea somewhat unnecessary. However, a holographic description of nature is
arguably very intriguing and unconventional!
Historical Overview – Motivation
In Plato’s “Allegory of the Cave”, people are chained and can only see their fuzzy shadows on
the cave walls cast by a fire. To them, the shadows represent the totality of their existence.
After several years of research, physicists studying the conjecture have
come to the conclusion that the validity and implications of the proposal go
beyond the context of the originally anticipated regime. It is now generally
referred to as gauge/gravity duality or holography. Recently, the rules of the
original conjecture were somewhat bent and a model of gauge/gravity duality
that is expected to be able to describe non-relativistic theories was proposed.
In the present work, the newly proposed extension is studied and an attempt is made to describe its behavior in different energy scales. To this end,
holographic renormalization techniques are employed. Chapters 1 to 3 present
the necessary tools and theoretical background. Holographic renormalization
is discussed in Chapter 4. In Chapter 5, we deal with basic principles and
certain aspects of non-relativistic holography. In the last chapter, the application of the holographic renormalization procedure in the non-relativistic case,
as well as related problems are presented.
3
1
Renormalization
The aim of this chapter is to introduce the so-called renormalization procedure.
Although a few ideas are described thoroughly, familiarity of the reader with
quantum field theory is assumed.
A very important problem that seemed to be inherent in quantum field
theories was the ultraviolet divergences that occurred. Therefore, in the earlier
days of quantum field theory (QFT) several theorists advocated the abandonment of that approach altogether. This was a reasonable conclusion at the
time, since the cancellation of ultraviolet divergences is essential if a theory is
to yield quantitative physical predictions.
We address some related issues by focusing on the method developed by
Kenneth Wilson [1]. It is not a bottom up approach that would introduce the
reader to all the basic techniques or provide sufficient background to deal with
related topics. However, it does provide a physical picture based on the scale
dependence of the theory’s parameters and it is essential to what is discussed
in the following chapters. Afterwards, we mention a few things about different
renormalization schemes.
1.1
Perturbative Renormalization - A Toy Model
We start by considering an unspecified theory which yields ultraviolet divergences. For simplicity we assume that it has only one free parameter. We
represent a physical quantity by F (x) which is calculated perturbatively in
terms of the free parameter λ0 , which is the coupling constant. If for example, we were talking about quantum electrodynamics (QED) F may represent
the cross section of a scattering process of an electron on a heavy nucleus. In
that case x would be the energy-momentum fourvector of the electron. So, we
assume that F (x) has the general form:
F (x) = λ0 + λ20 F1 (x) + λ30 F2 (x) + . . . ,
(1.1)
which up to a redefinition of F (x) corresponds to something we encouter in a
realistic field theory.
We further assume that the perturbative expansion is ill-defined in the
sense that Fi (x) are functions involving divergent quantities. For example, we
might encouter the following form
F1 (x) = α
ˆ
5
0
∞
dt
t+x
(1.2)
6
Renormalization
which shares common features with integrals in a QFT, since it is also logarithmically divergent in the upper limit. This integral in QFT would represent the
summation over virtual states and α (t + x)−1 would represent the probability
amplitude associated with each state.
The assumption made earlier–that we are talking about a one-parameter
theory–has as a consequence that only one measurement is enough fix the
value of λ0 , for example at the point x = µ.
It might seem redundant to parametrize the theory in terms of (the bare
parameter) λ0 , since it only seems helpful in intermediate calculations and
will be finally replaced with the physical, measured quantity F (µ). This freedom however is generic in physics. Moreover, in this particular case, there
is a subtlety since the singular behavior of the expansion dictates a singular
relationship between λ0 and F (µ). What we would do in general is follow the
following steps:
1. Decide on the Lagrangian that respects the required symmetries, locality
etc.
2. Perform calculations of physical processes using the bare quantities at
any required order in perturbation theory
3. Finally fix the parameter (or parameters for that matter) to reproduce
and/or predict experimental results.
A theory that is ill-defined however, suggests that F needs to be reparametrized
in terms of F (µ). This leads to what we might call the renormalizability hypothesis by which a reparametrization of the theory in terms of a physical
quantity instead of the bare parameters, is enough to render the perturbative expansion into a well-defined one. In other words, the problem of the
expansion is not the nature of the functions Fi (x) we used, but the choice of
the parameter we used to get the perturbative expansion. So, the physical
quantity F (x), should have a well-defined perturbative expansion once calculated in terms of the physical parameter F (µ). We are therefore led to define
the renormalized coupling constant (also referred to as the physical coupling
constant)
λR = F (µ) .
Unfortunately, we cannot use (1.1) because it is, by assumption, ill-defined.
We must somehow regularize the expansion to give it a well-defined meaning.
We accomplish that by introducing a new set of functions FΛ and Fi,Λ that
involve an new parameter Λ called the regulator.
Thus, we will work with the regularized functions FΛ and Fi,Λ which are
now finite for a finite value of Λ. We now have
FΛ (x, λ0 , Λ) = λ0 + λ20 F1,Λ (x) + λ30 F2,Λ (x) + . . .
(1.3)
1.1.
Perturbative Renormalization - A Toy Model
7
In order to regularize any Fi we introduce the cut-off for example in the
integral
ˆ Λ
dt
F1,Λ (x) = α
.
(1.4)
0 t+x
We can now obtain a well-defined perturbation series of FΛ in terms of the
physical coupling λR . If indeed, this expansion makes any physical sense, it
should do so even after we take the limit Λ → ∞ because it expresses a finite
physical quantity F (x) in terms of λR . So, we change F (x, λ0 ) to FΛ (x, λ0 , Λ)
and then rewrite FΛ in terms of λR and µ and then take the limit Λ → ∞
keeping λR and µ fixed. What we expect is
Λ→∞
F (x) = F (x, λR , Λ) = FΛ (x, λR , µ) .
We implement this order by order to see how it works exactly and what effect
it has on the perturbative expansion. At order λ0 we have
FΛ (x) = λ0 + O(λ20 ) .
So, we get
λ0 = λR + O(λ2R ) .
To second order λ20 we need to redefine λ0 in order to eliminate the divergence
of FΛ (x). We first expand λ0 as a power series in λR and obtain
λ0 = λ R + δ 2 λ + δ 3 λ + . . . ,
(1.5)
where δn λ ∼ O(λnR ). At λ2R order we get
FΛ = λR + δ2 λ + λ2R F1,Λ (x) + O(λ3R )
(1.6)
by using λ20 = λ2R + O(λ3R ). By using λR = F (µ) we obtain
δ2 λ = −λ2R F1,Λ (µ)
(1.7)
which diverges for Λ → ∞. In our case, using (1.4) we find
δ2 λ = −αλ2R
ˆ
0
Λ
dt
Λ+µ
= −αλ2R log
.
t+µ
µ
We substitute what we have so far back in the expansion to get
!
"
FΛ = λR + λ2R F1,Λ (x) − F1,Λ (µ) + O(λ3R ) .
This expression will be finite for all x at this order if the divergent part
(F1,Λ (x)) is exactly cancelled by that of F1Λ (µ), i.e.
F1,Λ (x) − F1,Λ (µ) is regular in x and µ for Λ → ∞ .
8
Renormalization
This translates into the fact that the divergent part of F1,Λ (x) must be independent of x and essentially a constant. We can now define the renormalized
F (x) as the limit of FΛ (x) for Λ → ∞. The above condition is satisfied for
the integral (1.2) and so
ˆ ∞
dt
2
F (x) = λR + α(µ − x)λR
+ O(λ3R ) .
(t + x)(t + µ)
0
We can now say that the theory is renormalized to this order. What we
“silently” did in (1.6) is an apparent “addition of a divergent” term δ2 λ to
cancel the divergence. These terms (which also occur in higher orders) are
called counterterms. They “appear” to absorb the infinite but unobservable
shifts between the bare parameters and the physical parameters. This mechanism is a generic: a divergence coming from the nth term of the perturbative
expansion is cancelled by the expansion in powers of λR of the n − 1 preceding
terms. It is noteworthy that this cancellation is possible only if the divergence
of F1,Λ (x) is independent of x, i.e. just a number. If that is not the case,
then F1,Λ (x) − F1,Λ (µ) will diverge for every x %= µ. This would entail going
through the renormalization procedure (at least) once again introducing (at
least) one more independent coupling constant, which does not agree with the
fact that our theory is defined to have only one free parameter.
The analysis follows through for higher orders in perturbation theory. So
we would have
FΛ (x) = λR + δ2 λ + δ3 λ + (λ2R + 2λR δ2 λ)F1,Λ (x) + λ3R F2,Λ (x) + O(λ4R ) ,
where we have used λ30 = λ3R +O(λ4R ) and λ20 = λ2R +2λR δ2 λ+O(λ4R ). Imposing
λR = F (µ) once again, we obtain
!
"2
δ3 λ = 2λ3R F1,Λ (µ)
Substituting back, we obtain
!
− λ3R F2,Λ (µ) .
"
#
FΛ (x) = λR + λ2R F1,Λ (x) − F1,Λ (µ) + λ3R F2,Λ (x) − F2,Λ (µ)−
!
"$
2F1,Λ (µ) F1,Λ (x) − F1,Λ (µ)
To eliminate the divergence, we require
!
"
F2,Λ (x) − F2,Λ (µ) − 2F1,Λ (µ) F1,Λ (x) − F1,Λ (µ)
+ O(λ4R ) .
(1.8)
to be regular in x and µ when Λ → ∞. The new feature now is that we also
have F1,Λ in our constraint. For convenience we split up the Fi,Λ ’s in a regular
and a singular part for this limit
s
r
Fi,Λ (x) = Fi,Λ
(x) + Fi,Λ
(x) .
1.1.
Perturbative Renormalization - A Toy Model
9
This decomposition is not unique since adding anything to infinity yields ins (x)’s are defined up to a regular part. It is convenient
finity again. So, the Fi,Λ
to make the choice
Λ→∞
s
s
F1,Λ
(x) − F1,Λ
(µ) −→ 0
(1.9)
which is also implied by the constraint in the analysis for the expansion to the
second order in λ. This choice is always possible if the previous constraint is
fulfilled[2]. This means that the divergent part of F1,Λ is independent of x.
s and choose
We can go a step further and impose a tighter constraint to F1,Λ
it to be completely independent of x for every Λ, because we can tune the
regular part of F1,Λ . So, we define
s
F1,Λ
(x) = f1 (Λ) .
(1.10)
For our integral (1.2) we can choose
r
f1 (Λ) = α log Λ and F1,Λ
(x) = α log
%
Λ+x
Λx
&
.
We now substitute back in our constraint (1.8) and obtain
'
(
s
s
r
r
F2,Λ
(x) − F2,Λ
(µ) − 2f1 (Λ) F1,Λ
(x) − F1,Λ
(µ)
which can be rewritten as
'
(
'
Λ→∞
−→ 0
(
s
r
s
r
F2,Λ
(x) − 2f1 (Λ)F1,Λ
(x) − F2,Λ
(µ) − 2f1 (Λ)F1,Λ
(µ)
Λ→∞
−→ 0 .
s
s −
The structure is the same as (1.9) up to the replacement F1,Λ
→ F2,Λ
r
2f1 (Λ)F1,Λ and can therefore have the same solution as (1.10). We have:
s
r
F2,Λ
(x) = 2f1 (Λ)F1,Λ
(x) + f2 (Λ)
where f2 (Λ) is any function of Λ independent of x. Apparently, the divergent
s . This dependence however is
part of F2,Λ does depends on x, unlike F1,Λ
determined by the first order of the perturbative expansion. The δ2 λ term
used to remove the O(λ20 ) divergence, has produced an x-dependent divergent
term at order λ3R , namely 2λR δ2 λF1,Λ (x). This kind of dependence is generic
in renormalization. The counterterms that remove divergences at a given
order produce divergences in higher orders. If the theory is renormalizable,
these divergences contribute to the cancellation of divergences that appear
at higher order. In this sense, this procedure suggests a precise structure
of the divergent parts of the successive terms of the perturbative expansion.
At nth order, the singular part of Fn,Λ involves x-dependent terms entirely
determined by the preceding terms plus one new that is x-dependent. In our
case, we find
&
%
Λ+x
s
2
+ f2 (Λ) .
F2,Λ (x) = 2α log Λ log
Λx
10
Renormalization
Going to higher orders makes it increasingly technical without adding to
the circle of ideas already presented. As a rule of thumb, renormalized perturbation theory is technically easier, especially for multiloop diagrams (higher
order expansion).
What we essentially did so far is the following. We just reparametrized our
theory in terms of the physical quantity λR . By renormalizing F we pay the
price of letting λ0 go to infinity for the limit Λ → ∞ (if we plug (1.7) in (1.5)).
But this is not a problem because λ0 is not a measurable physical parameter. It
is merely something helpful for intermediate calculations. The interpretation
of the procedure we followed however is rather obscure. It seems like we first
introduced unphysical (bare) quantities that make everything infinite, and
then rewrite everything in a way that it looks like we “added” other divergent
quantities (counterterms) to compensate for the original divergences.
Nonetheless, the program works! In a realistic, renormalizable theory, we
can follow the procedure outlined in our toy model and get sensible results that
agree with experiments. Until the physical interpretation of the renormalization procedure was developed by Ken Wilson many people felt uneasy about
renormalization. At first, it seems like it is nothing more that a well-organized
mathematical trick to hide what is not well understood. The physical picture
that we will eventually discuss in the following is that of renormalization group
flows in the space of theories.
1.2
Wilsonian Renormalization Theory
To discuss Wilson’s analysis it suffices to work in φ4 theory. This will provide
the basic qualitative results of the renormalization group (RG) program. We
will further make this discussion more intelligible by using a sharp momentum
cutoff instead of the method of dimensional regularization which is briefly
described in Appendix A. This is also more closely related to the RG procedure
in the context of the gauge/gravity duality which is discussed in Chapter 4.
The starting point is the construction of Green’s functions of the φ4 theory
in terms of a functional integral representation of the generating functional
Z[J]
ˆ
Z[J] =
Λ
Dφ ei
´
[L+Jφ]
.
We then impose a sharp
cutoff Λ by integrating only over the field
´ ultraviolet
dd k ikx
configurations φ(x) = (2π)d e φ(k) such that φ(k) = 0 for |k| > Λ. This
amounts to taking into account the influence of quantum fluctuations at very
short distances (L = 1/Λ) or equivalently very large momenta. But to ensure
that large momenta are controlled we have to ensure that we are working
in Euclidean space. This is crucial. If we remain in Minkowski space, the
lightlike components of k can be large while k 2 remains very small. In order
to obtain Euclidean momenta we perform a Wick rotation. This also makes
1.2.
Wilsonian Renormalization Theory
11
this treatment relevant to statistical mechanics∗ . Incidentally, when the need
to rescale time and space differently as done in condensed matter physics (e.g.
the surface growth problem), the dynamical exponent z is employed. This
measures the difference in the rescaling and is discussed in section 5.1.
The next step is to to carry out the integration over the high-momentum
degrees of freedom φ. We set J = 0 and we let Λ → Λ − δΛ (with δΛ > 0). We
then write φ = φs + φf (let us use s for “slow” and f for “fast”). The fields φs
and φf are defined in such a way that the Fourier components are non zero
for |k| ≤ (Λ − δΛ) and (Λ − δΛ) ≤ |k| ≤ Λ respectively. For convenience we
introduce a real number b < 1 and rewrite Λ − δΛ = bΛ.
The generating functional for the φ4 theory then becomes
)
*
1
1
(∂µ φs + ∂µ φf )2 + (φs + φf )2
2
2
+,
λ
4
+
(φs + φf )
4!
ˆ
ˆ
ˆ
)
*
´
1
1
= Dφs e− L(φs ) Dφf exp − dd x
(∂µ φf )2 + φ2f
2
2
%
&+,
1 3
1 2 2 1 3
1
+λ
φs φf + φs φf + φs φf + φ4f
(1.11)
6
4
6
4
Z=
ˆ
Dφs
ˆ
Dφf exp −
ˆ
d
d x
It should be noted that terms quadratic in φs φf vanish, since Fourier
components of different wavelengths are orthogonal. Obviously, in the final
expression all terms independent of φf are gathered in L(φs ).
After an integration over φf the result should be of the form
ˆ
% ˆ
&
d
Z = [Dφ]bΛ exp − d x Leff .
Here Leff has only the Fourier components of φ(k) with |k| < bΛ. As it turns
out, the effective Lagrangian density is equivalent to the original with some
added corrections proportional to powers of λ. These corrections emerge so
that they will compensate for the removal of the large momentum Fourier
components φf . This is achieved by the interactions among the remaining
φ(k) that were previously mediated by fluctuations of the φf .
We now
wish to compare the '
original functional
integral Z[J = 0] =
(
´
´
´
´ d
i
L
and Z = [Dφ]bΛ exp − d x Leff . We will treat the terms
Λ Dφ e
after the first one (the kinetic term) as small perturbations. This is a valid
approximation as long as the coupling constants are small. In order to do this
we introduce rescaled distances and momenta in the latter
k % = k/b
and
4
x% = xb ,
(1.12)
The Euclidean functional integral for φ theory has precisely the same form as the
continuum description of the statistical mechanics of a magnet, where the field φ(x) is
interpreted as the fluctuating spin field s(x).
∗
12
Renormalization
so that the integral is carried out over |k % | < Λ. As we already mentioned the
Leff is the original Lagrangian density with added corrections which are essentially a sum of connected diagrams. This means that we can write (dropping
the subscript of the field)
ˆ
d x Leff =
d
ˆ
dd x
*
(
1
1' 2
m + ∆m2 φ2
(1 + ∆Z) (∂µ φ)2 +
2
2
+
1
+ (λ + ∆λ) φ4 + ∆C (∂µ φ)4 + ∆Dφ6 + . . . .
4!
Using the rescaled variable x% we obtain
ˆ
ˆ
*
'
(2
(
1
1' 2
d
d x Leff = dd x b−d
m + ∆m2 φ2
(1 + ∆Z) b2 ∂µ% φ +
2
2
+
1
+ (λ + ∆λ) φ4 + ∆Cb4 (∂µ φ)4 + ∆Dφ6 + . . . ,
4!
where Z is the so-called field-strength renormalization.
The original functional integral (1.11) gives rise to the propagator
´
! ´
"
Dφf exp − L0 φf (k)φf (p)
´
! ´
"
φf (k)φf (p) =
Dφf exp − L0
1
=
(2π)d δ (d) (k + p)Θ(k) ,
k2
where
Θ(k) =
-
1
0
if bΛ < |k| < Λ
otherwise.
It is reasonable to demand that the new functional should lead to the same
propagator. This is done by rescaling the field according to
.
/1/2
φ% = b2−d (1 + ∆Z)
φ.
The rescaling leads to a transformation of the perturbations:
ˆ
ˆ
* '
'
(4
1 % % (2 1 %2 %2
1
d
d
d x Leff = d x
∂µ φ + m φ + λ% φ4 + ∆C % ∂µ% φ%
2
2
4!
+
+∆D φ + . . . .
% %6
Now, the new parameters of the Lagrangian are
m%2 =
'
(
m2 + ∆m2 (1 + ∆Z)−1 b−2
λ% = (λ + ∆λ) (1 + ∆Z)−2 bd−4
C % = (C + ∆C) (1 + ∆Z)−2 bd
D% = (D + ∆D) (1 + ∆Z)−3 b2d−6 .
(1.13)
1.3.
Renormalization Group Flows
13
The combination of integrating out of the high momentum degrees of freedom and the rescaling (1.12) we can viewed as an operation acting on the
Lagrangian as a transformation. This procedure can be repeated and integrate over another shell of momentum space. By doing so, another set of
transformations is inevitably introduced, analogous to (1.13). In the limit
where the parameter b is close to 1 the transformation becomes a continuous
one. In this sense, the result of this procedure we view this integration of a
field theory as a trajectory or a flow in the space of all possible Lagrangians
and here lies the core of the renormalization in the Wilsonian sense. The term
renormalization group is attributed due to the continuous generated transformations of Lagrangians. This of course does not constitute a group in any
formal way, since the integration of degrees of freedom is not invertible.
1.3
Renormalization Group Flows
Let us consider a simple case where the Lagrangian is in the vicinity of the
point where all the perturbations vanish. This means that we are close to the
point where m2 , λ, C, D. . . are equal to zero. The transformations will leave
this point unchanged. So, we say that the free-field Lagrangian
1
L0 = ∂µ φ∂ µ φ
2
is a fixed point of the renormalization group transformation.
Keeping only terms that are linear in perturbations gives a simple transformation law:
m%2 = m2 b−2 ,
λ% = λbd−4 ,
C % = Cbd ,
D% = Db2d−6 .
(1.14)
Since b < 1, parameters multiplied by a negative power of b will grow, while
the others will decay. The growing coefficients of the Lagrangian will carry it
away from L0 .
The various terms in Leff can be thought of as a set of local operators
that can be added as perturbations to L0 . As we are interested in longer
distance scales (or lower energies) couplings are affected in different ways.
The operators with coefficients that grow during the iterative procedure of
transformation are called relevant. Those that decay are called irrelevant and
those that are multiplied by b0 are called marginal. For example, the mass
term φ2 is always relevant but the φ4 term is relevant only if d < 4. In the
case of d = 4 the φ4 term is marginal. The general formula that determines
the transformation of an operator with N powers of φ and M derivatives is:
%
CN,M
= bN (d/2−1)+M −d CN,M .
From a Wilsonian point of view, any quantum field theory is defined fundamentally with a cutoff Λ that has some physical significance. For statistical
14
Renormalization
mechanics this is translated as the inverse atomic spacing. In Quantum Electrodynamics and other relevant to high energy physics theories, the cutoff
is associated with some fundamental graininess of spacetime. What all this
means is that no matter what the Lagrangian looks like at its fundamental
scale, as long as the couplings are sufficiently weak, it must be described at
the energies of our experiments by a renormalizable effective Lagrangian, Leff .
As an illustration we turn again to the renormalization group flows near
L0 for the case of φ4 theory. We consider three distinct cases d > 4, d = 4 and
d < 4 as shown in Fig. 1.1. For d > 4 the only relevant operators is the mass
term since it increases near the point of L0 . Meanwhile, the φ4 term and all
other higher order interactions decay.
m2
d>4
d=4
m2
λ
λ
m2
d<4
λ
Fig. 1.1: RG flows near the free-field fixed point [3].
If we consider the d = 4 case the transformation law (1.14) does not suffice
to decide whether the φ4 term is relevant or irrelevant. We have to use the
complete set of transformations (1.13). We find the transformation
λ% = λ −
'
(
3λ2
−1
log
b
16π 2
1.3.
Renormalization Group Flows
15
which suggests that λ slowly decreases as we integrate out high momentum
degrees of freedom.
For the final case of d < 4, λ becomes a relevant parameter. This means
that the theory flows away from the free theory L0 as we integrate out degrees
of freedom. In other words, in large distances the φ4 interaction becomes
increasingly important. But due to the nonlinear corrections that occur when
λ becomes large at one-loop, we find for d < 4 that
0
3λ2
1
bd−4 − 1 d−4 d−4
λ = λ−
Λ
b
,
(4π)d/2 Γ (d/2) 4 − d
%
What this implies is that there is a value of λ at which the decrease due to
the nonlinear effect compensates the increase due to rescaling. At this point,
λ remains unchanged as we integrate out degrees of freedom. This leads to a
second fixed point of the renormalization group flow. By taking the limit of
d → 4 the new fixed point merges with the free field theory fixed point and
shares the property that the mass parameter m2 is increased by iteration. So,
the mass operator will be a relevant operator near the new fixed point and the
form of the flow “degenerates” to the form of the d = 4 case. In this region
(d < 4), the case of d = 2 is especially interesting: all the operators (of any
power) become relevant.
In condensed matter physics, for a given scalar field theory, the dimensions
d at which most relevant interactions become marginal is known as the critical
dimension. For example, in the case of the φ6 theory, the critical dimension
is 3.
In a quantum field theory in a arbitrary scale µ the coupling constant
evolves according to the renormalization group flow equation
µ
dg
= β(g) .
dµ
For a theory with several (N ) coupling constants we can write
dgi
= βi (g1 , . . . , gN )
dt
upon defining t ≡ log(µ/µ0 ). It is also known as the Gell-Mann–Low equation.
In this picture, we think of (g1 , . . . , gN ) as the coordinate of a particle in
N -dimensional space, t as time, and βi (g1 , . . . , gN ) as a position dependent
velocity field. The idea is to study how the particle moves or flows as we
tamper with µ and t. Obviously, the point where β vanishes is of particular
∗ ) is called the aforementioned fixed point. We
interest. This point (g1∗ , . . . , gN
can distinguish three classes:
1. Stable fixed points, whose scaling fields are all irrelevant, or at worst
marginal. These points define in condensed matter physics what we
16
Renormalization
might call “stable phases of matter”. When a system is released somewhere in the parameter space surrounding any of these attractors, it will
scale toowards the fixed point and eventually sit there. In other words,
when the system is studied in larger and larger distance scales it will
resemble the infinitely correlated self-similar fixed point configuration.
2. Unstable fixed points, whose scaling fields are all relevant. These fixed
point represent the ancient old idea of never being able to get there! The
conditions near the fixed point will force the system to flow away from
it. Despite the lack of realizable forms of matter that correspond to this
situation, they are still important as they “orient” the global RG flow
of the system.
3. Generic class of fixed points with both relevant and irrelevant scaling
fields. These points present the interesting association to phase transitions. In condensed matter at fixed points the intrinsic length ξ is either
zero or infinite. The first case is not interesting. The latter however,
a diverging correlation length ξ → ∞, is an indicator of a second-order
phase transition.
As a general remark, we mention again that once the coupling constants are
fine-tuned to a fixed point, the system no longer changes under subsequent
RG transformations. In particular, it remains invariant under the change of
space or time scale associated with the transformation. So, they look the same
no matter how closely we observe them. Systems that also share this property
are fractals (Fig. 1.2).
Field theories that are insensitive to this scaling transformation, that have
zero β functions, are often referred to as conformal field theories. There are
also some other considerations involved that have to do with symmetries, but
these will be addressed later† .
If we want to study behaviour of a theory at high energies we need to find
all its stable points under the renormalization group flow. Some couplings
may flow toward larger values while other are flowing toward zero.
In practice however, this is difficult to implement since we have no way of
calculating the functions βi (g). What is more, the point g ∗ can be quite far
away from zero and this is known as a strong coupling fixed point. In this
case perturbation theory and Feynman diagrams break down and are of no
use in determining the properties of the theory there. In fact the fixed point
structure of very few theories is known.
Strictly speaking the conformal group is a much larger group that includes scaling
transformations. Be that as it may, cases of quantum field theories that are scale invariant
and not conformally invariant are extremely rare (see [4] for a counterexample). Therefore,
the terms scale-invariant and conformally-invariant are customarily used interchangeably.
†
1.4.
The Callan-Symanzik Equation
Fig. 1.2: This is an Apollonian packing, in which the triangular gap between three
touching circles is filled with another circle. That leaves three more gaps, which
are filled by three smaller circles, etc. Similar insensitivity to zooming in or out
characterizes physical systems at fixed point.
1.4
The Callan-Symanzik Equation
Going back to the φ4 theory let us now suppose that we want to define the
same theory at a different scale µ% . This means that we are looking at a theory
whose bare Green’s functions
(Ω| T φ0 (x1 )φ0 (x2 ) . . . φ0 (xn ) |Ω)
are given by the same functions of the bare coupling constant λ0 and the cutoff
Λ. Here |Ω) represents the ground state. The scale dependence enters when
the cutoff dependence is removed by rescaling the fields and eliminating λ0 in
favor of the renormalized coupling λ. Green’s functions remain the same up
to a rescaling by powers of the field strength renormalization: Z −n/2 .
Let us suppose that we shift the scale by δµ. This leads to
µ →µ + δµ
λ →λ + δλ
φ →(1 + δη)φ .
The induced renormalized Green’s function is
G(n) → (1 + nδη)G(n) .
17
18
Renormalization
Treating G(n) as a functional of µ and λ we write
dG(n) =
∂G(n)
∂G(n)
δµ +
δλ = n∂ηG(n) .
∂µ
∂λ
Conventionally though, what is used is
β≡
µ
δλ
δµ
γ≡−
µ
δη .
δµ
This is the same β function that we mentioned in a more abstract framework
in the discussion of the RG flows. Substituting back and multiplying by µ/δµ
we obtain
*
+
∂
∂
µ
+ β(λ)
+ nγ(λ) G(n) (x1 , . . . , xn ; µ, λ) = 0 ,
∂µ
∂λ
which is the so-called Callan-Symanzik equation. The parameters β and γ
are the same for every n and must be independent of xi . Since G(n) is renormalized, β and γ cannot depend Λ and therefore, by dimensional analysis,
these functions cannot depend on µ. They are functions of the dimensionless
variable λ only! So, Green’s functions of massless φ4 theory must satisfy the
Callan-Symanzik equation. What it tells us is that there exist two universal
functions β(λ) and γ(λ), related to the shifts in the coupling constant and
field strength, that compensate for the shift in the renormalization scale µ.
The generalization to other massless theories with dimensionless couplings is
straightforward. We mention for example QED at zero electron mass gives
*
µ
+
∂
∂
+ β(e) + nγ2 (e) + mγ3 (e) G(n,m) (x1 , . . . , xn ; µ, e) = 0 ,
∂µ
∂e
where n and m are the number of electron and photon fields respectively in
the Green’s function and γ2 and γ3 are the rescaling functions of the electron
and photon fields.
1.5
Renormalization Schemes
Several ways of renormalizing a theory have been developed by field theorists
throughout the years. We will not discuss them thoroughly. We will only
describe one of them and make some comments on scheme dependence.
The most popular renormalization schemes are the Minimal Subtraction
(MS) and the related modified minimal subtraction (MS), the Dimensional
Regularization (DR) and the modified version of that (DR). We directly
present the rules for the MS scheme:
1. Dimensional regularization to control ultraviolet divergences
1.5.
Renormalization Schemes
19
2. Identify the µ parameter of dimensional regularization
D
d4 p
4−D d p
→
µ
(2π)4
(2π)D
with the energy scale!of the"renormalization group. This identification
set the logarithm log µ2 /µ20 to zero.
3. The ultraviolet divergence of a L-loop amplitude is a Lth -degree polynomial in 1/$:
%
&
AL AL−1
A1
2
+
+
.
.
.
+
+
f
(p
)
(1.15)
+L
+L−1
+1
for some constants AL , . . . , A1 and with f (p2 ) being a finite function of
p2 . To cancel the divergence we need a L-loop order counterterm
δLZ
=g
2L
%
AL AL−1
A1
+ L−1 + . . . + 1 + A0
L
+
+
+
&
,
where the coefficients are now completely determined by the ultraviolet
divergence of the L-loop diagrams, but the finite term A0 is not. Its value
is determined from the renormalization scheme we use for the amplitude
1.15 and not from the divergence.
In the MS scheme there are no conditions imposed on the amplitudes. Instead,
we set A0 = 0. Similarly, the finite parts of all the other counterterms are
set to zero. The name comes from the fact that the counterterms simply
subtract the pole at + = 0. The finite part of the amplitude is whatever the
loop diagrams produce and the counterterms do not mess with it. In the
similar modified minimal subtraction scheme MS, one absorbs the divergent
part plus a universal constant (which always arises along with the divergence
in Feynman diagram calculations) into the counterterms.
The DR and DR schemes are more often used in supersymmetric theories.
The schemes are similar to MS and MS schemes with a different approach
to dimensional regularization (Appendix A) called dimensional reduction. In
this case, all momenta live in D = 4 − 2+ dimensions but the vector fields
maintain all four components. Such a reduced four-dimensional vector field
comprises one species of a D-dimensional vector plus 2+ species of scalar with
the same mass and charge. Unlike the original dimensional regularization, the
dimensional reduction respects supersymmetry and that is the main difference
between the two.
Scheme (in)dependence
The renormalization scheme is expected to modify the coupling constant, and
in fact it does. So, for the same energy we would end up with a slightly
20
Renormalization
different running coupling λ% (µ0 ) %= λ(µ0 ). The differences usually appear at
one loop and are due to quantum corrections
λ% (µ0 ) = λ(µ0 ) + O(λ2 (µ0 )) .
The beta function however turns out to be independent of the renormalization
scheme up to three-loop level. We can prove this as follows.
We write the coupling constant as a power series
λ% (µ0 ) = λ(µ0 ) + c1 λ2 (µ0 ) + c2 λ3 + . . .
(1.16)
with some constants c1 , c2 , . . . The inverse coupling would be
(
'
1
1
2
λ(µ0 ) + . . .
=
−
c
+
c
−
c
2
1
1
λ% (µ0 )
λ(µ0 )
The inverse coupling depends on energy according to
d
1
−1 dλ(µ0 )
−1
= 2
= 2
β(λ(µ0 )) .
d log E λ(µ0 )
λ (µ0 ) d log µ0
λ (µ0 )
We can write the beta function in power series
β(λ) = b1 λ2 + b2 λ3 + b3 λ4 + . . .
and use it to get
d
1
= −b1 − b2 λ(µ0 ) − b3 λ2 (µ0 ) − . . .
d log(µ0 ) λ(µ0 )
Let us assume that from a different renormalization scheme we obtained the
β-function
β % (λ% ) = b%1 λ%2 + b%2 λ%3 + b%3 λ%4 + . . .
and consequently
d
1
= −b%1 − b%2 λ% (µ0 ) − b%3 λ%2 (µ0 ) − . . .
%
d log(µ0 ) λ (µ0 )
(1.17)
We now differentiate both sides of the coupling constant expansion (1.16) and
obtain
*
+'
'
(
d
1
−1
2
= 2
+ c1 − c2 + . . . −b1 λ2 (µ0 ) − b2 λ3 (µ0 )
d log(µ0 ) λ% (µ0 )
λ (E)
.
'
= −b1 − b2 λ(µ0 ) − b3 − b1 c21 − c2
.
(/
'
λ2 (µ0 ) − . . .
= −b1 − b2 λ% (µ0 ) − b3 − −b2 c1 − b1 c21 − c2
(
−b3 λ4 (µ0 ) − . . .
(/
λ%2 (µ0 ) − . . .
By comparison with (1.17) we see that b%1 = b1 and b%2 = b2 but b%3 %= b3
and we can guess that higher order coefficients are also different. A similar
theorem applies to theories with more coupling constants for the three-loop
and higher-order terms. Of course, physical observables are unaffected by the
renormalization scheme used.
2
Action Principle and
Geometry
As a universal strategy in physics, the relation of the Hamiltonian and the
generator of time translations is used to determine the time evolution of a
physical system. The Hamiltonian is also directly related to the total energy
of a physical system. It is therefore useful to have a Hamiltonian formulation
of a physical theory. However, in General Relativity, this turns out to be a
non-trivial task. However, it was done in a satisfactory manner in the pivotal
paper by Arnowitt, Deser, Misner [5] (ADM). They provided a consistent
way of discussing energy in curved spacetime, yielding positive values and
obeying fundamental conservation principles. This also opened the way for
positive energy theorems [6] (an interesting alternative proof can be found
in [7]) This solidified the fact that Minkowski space is stable. As we will
see (Chapter 2.2), the coordinate invariance underlying the theory creates an
analogous problem to gauge invariance in electromagnetic theory. This makes
the required breakup of spacetime into space and time more subtle than in,
for example, classical field theory.
Eventually, we will want to use the Hamilton-Jacobi theory. Since this
can be viewed in a more elemental level and is not always treated in a course
on (classical) mechanics nor (classical) field theory, we will discuss this first.
We will also have a chance to present some notions that are useful to what is
discussed in following chapters. Afterwards, we will present the Hamiltonian
formulation of General Relativity and therein, discuss Hamilton constraints.
2.1
Hamilton-Jacobi Theory
We start by describing the idea of Hamilton and Jacobi in the context of
classical mechanics.
We define the action for a trajectory as a functional of the independent
varables qi and pi and the Hamiltonian
S=
ˆt1
t0
dt (pi (t)dqi (t) − H (qi (t), pi (t), t)) .
(2.1)
According to the modified Hamilton’s principle the true path extremizes the
action for whatever independent variation of qi (t) and pi (t), for which the
initial and final positions qi (t1 ) , qi (t2 ) are fixed. The difference between the
21
22
Action Principle and Geometry
modified and the original Hamilton’s principle is that we treat both position
and momentum as independent variables. In order to extremize the action we
vary both qi (t) and pi (t), not just qi (t).
Next, we make the observation that, even in the Lagrangian formulation,
adding a total time derivative of a function F (q, p, t) will not affect the true
path
S=
ˆt1
t0
dt
2%
i
&
dF (q, p, t)
pi (t)dqi (t) − H (qi (t), pi (t), t) +
.
dt
(2.2)
Indeed, the true path is the same for both (2.1) and (2.2). This observation
is important because it allows us to introduce a class of coordinate transformations
Qi = Qi (q, p, t) , Pi = Pi (q, p, t)
with the all-important property that the true path satisfies Hamilton’s equations in the new coordinates:
Q̇i =
∂K
∂K
, Ṗi = −
,
∂Pi
∂Qi
(2.3)
where K is the Hamiltonian in the new coordinates that must be determined.
The class of transformations that satisfy (2.3) are called canonical transformations. It is important to note that if the transformation satisfies:
pi q̇i − H (qi , pi , t) = Pi Q̇i − K (Qi , Pi , t) +
dF
dt
(2.4)
then the true path that extremizes the action
ˆt1
S1 =
dt
t0
2
i
(Pi dQi − K(Q, P, t))
satisfies (2.3) and also extremizes
S=
ˆt1
dt
t0
2
i
(pi (t)dqi (t) − H (qi (t), pi (t), t))
by virtue of (2.4). Therefore, equations (2.3) describe the true path that
satisfies Hamilton’s equations q̇i = ∂H/∂pi , ṗi = −∂H/∂qi but now in the
new coordinates Q, P . In other words, canonical transformations must satisfy
the form of (2.4).
Furthermore, (2.4) can be rewritten as
dF =
2
i
(pi dqi − Pi dQi ) + (K − H) dt ,
(2.5)
2.1.
Hamilton-Jacobi Theory
23
in which case we can choose a function F1 (q, Q, t). Thus, for the example
where F depends only on q, Q and possibly time t, we have
dF1 =
2 % ∂F1
∂qi
i
dqi +
&
∂F1
∂F1
dQi +
dt .
∂Qi
∂t
(2.6)
Then, the canonical transformation (q, p) → (Q, P ) is indirectly defined by
the relations
pi =
∂F1
,
∂qi
Pi = −
∂F1
.
∂Qi
(2.7)
This is so because we can determine the new coordinates as functions of the
old ones Qi = Qi (q, p, t), Pi = Pi (q, p, t) by solving the algebraic equations
(2.7). Comparing (2.5) and (2.6) we can see that the Hamiltonian in the new
coordinates is
∂F1
K=H+
.
∂t
The radical idea of Hamilton and Jacobi was to make a canonical transformation in which the Hamiltonian K is zero. If a transformation like that
can be determined then the initial dynamical problem is immediately solved
in the new coordinates since the equations of motion become
Q̇i = 0 ,
Ṗi = 0 .
H(q, p, t) +
∂F
=0
∂t
We immediately see that
through which one can define F (q, p, t). But we also have pi = ∂F/∂qi . Therefore, the differential equation that F needs to satisfy can be written as
%
&
∂F
∂F
+ H qi ,
,t = 0.
∂t
∂qi
(2.8)
Equation (2.8) that determines the required transformation is called the
Hamilton-Jacobi equation. Solving the Hamilton-Jacobi equation is equivalent
to solving Hamilton’s equations.
As a partial differential equation it can have more than one solution. As an
example we can work out one solution of the Hamilton-Jacobi for the simple
case of the free particle Hamiltonian
H=
If we take
F (q, p, t) =
p2
.
2m
(q − Q)2
2m(t − t% )
(2.9)
24
Action Principle and Geometry
as a solution, we can see that this function satisfies the Hamilton-Jacobi equation:
%
&
∂F
1 ∂F 2
+
= 0.
∂t
2m ∂q
Therefore, we know that it will provide the required transformation. We can
now discuss the physical interpretation of F . We can recognize in (2.9) that
it describes the movement of a particle that was at the time t% at the position
Q and at time t at the position q. This is more than just a coincidence.
Let us define the function S(q1 , q2 , t1 , t2 ) as the action that emerges from
the true path that goes through q1 (t1 ) and q2 (t2 ). The function
S(q1 , q2 , t1 , t2 ) =
ˆt1
t2
dt (pq̇ − H) =
ˆq2
q1
dq p −
ˆt2
dt H
t1
is no longer a functional, but merely a function in position space that depends
only on the initial and final positions and times, the connection of which is
achieved automatically via the true path. A small spacetime variation of the
initial and final points will result in the variation of the function action
dS = p2 dq2 − p1 dq1 − H2 dt2 + H1 dt1
(2.10)
where p1,2 and H1,2 are the values of momenta and the Hamiltonian at the
endpoints. From (2.10) we can deduce
p2 =
∂S
,
∂q2
p1 = −
∂S
∂q1
and
H2 = −
∂S
,
∂t2
H1 =
∂S
.
∂t1
Therefore, the function S(q, Q, t, t% ) that corresponds to the function of the
true path from Q at time t% to q at time t is a solution of the Hamilton-Jacobi
equation
%
&
∂S
∂S
+ H qi ,
,t = 0.
∂t
∂qi
This makes the function S(q, Q, t, t% ) the desired function F (q, Q, t) that generates the canonical transformation from (q, p) to (Q, P ) and makes the Hamiltonian expressed in the coordinates Q, P to equal zero.
However, determining S(q, Q, t, t% ) requires the knowledge of the true path,
and therefore the idea of Hamilton-Jacobi is not a panacea for the solution
of the dynamical equations. Nonetheless, when the Hamiltonian as it appears in (2.8) can be split in terms that depend on separate variables the
Hamilton-Jacobi equation can provide a fast mathematical solution to the
physical problem.
2.2.
2.2
Electromagnetism and the Hamiltonian
25
Electromagnetism and the Hamiltonian
Before embarking in formulating General Relativity in terms of the Hamiltonian, it is useful to see similar subtleties and problems that arise in electromagnetism in Minkowski spacetime, for similar reasons.
Although it might seem straightforward to apply the standard procedure
to obtain the Hamiltonian formulation in the classical electromagnetic field in
Minkowski spacetime, it is actually a bit more involved than that. Let us see
how this comes about.
We may start by writing the Lagrangian density in ordinary 3-dimensional
vector notation:
LEM
%
1 ,̇ ,
A + ∇V
=
2
& %
,̇ + ∇V
,
· A
&
−
( '
(
1 ',
, · ∇
, ×A
, .
∇×A
2
, is
The momentum conjugate to A
,̇ + ∇V
, ≡ −E
,.
,π = A
However, it is immediately obvious that there is a problem: V̇ does not appear in LEM . This means that the momentum πV conjugate to V vanishes
identically:
πV = 0 .
On the other hand, the Hamiltonian density H is defined as
H(q, π) = π q̇ − L .
This means that we cannot obtain an invertible relation between π and q̇.
In other words, we cannot eliminate q̇ in favor of π and q. This leads to
a breakdown of our general prescription for obtaining a Hamiltonian. This
, which prohibits getting
is directly related to the arbitrariness of gauge in A
deterministic dynamics for the gauge field of the form
δH
δπ
δH
π̇ ≡ −
.
δq
q̇ ≡
(2.11a)
(2.11b)
Fortunately, there is a way to resolve this. We can think of V as not a
dynamical variable since πV vanishes identically. So, we define
,̇ − LEM
HEM = ,π · A
1
1, ,
,
=
,π · ,π + B
· B − ,π · ∇V
2
2
1
1, ,
, · ,π − ∇
, · (V ,π ) .
=
,π · ,π + B
·B+V∇
2
2
26
Action Principle and Geometry
The total divergence term at the end contributes only as a boundary term in
the action and will vanish in the limit as the boundary goes to infinity for the
, The main point is
usual asymptotic conditions imposed on V and ,π = −E.
,
that we view HEM as a functional of A and ,π , with V playing the effective
role of a Lagrange multiplier. This means adding the equation
δHEM
=0
δV
,̇ and ,̇π ∗ . The newly added equation essentially
to (2.11a) and (2.11b) for A
suggests
, ·E
, =0
∇
(2.12)
and (2.11a), (2.11b) yield
,̇ = −E
, − ∇V
,
A
'
(
, ×A
,
,̇ = −∇
, × ∇
,̇π = −E
(2.13a)
(2.13b)
and all of them are equivalent to Maxwell’s equations.
This result can be viewed as obtaining the constraint (2.12) and the evolution equations (2.11). As expected the value of HEM for a solution of Maxwell’s
equations will be proportional to the total energy of the electromagnetic field.
This type of Hamiltonian formulation where a non-dynamical variable appears
in H and is effectively a Lagrange multiplier is called constrained Hamiltonian
formulation. It is a general feature, expected to arise, in a theory where the
field variables have a gauge arbitrariness.
2.3
The ADM Formalism
Discussing the ADM [5] formulation of the dynamics of geometry boils down
to carefully formulating General Relativity as a field theory, using the Hamiltonian instead of the Lagrangian. A Hamiltonian formulation of a field theory
requires a breakup of spacetime into space and time. This is subtle in the
case of General Relativity and should be done with some caution. The reason
for this, as we already mentioned, is that coordinate invariance underlying
the theory creates analogous problems to gauge invariance in electromagnetic
theory. Comprehensive discussions on this topic can be found in [8] and [9].
Here, we will primarily focus on aspects related to Hamilton constraints in
the presence of gravity.
The idea of slicing spacetime into a one-parameter family of spacelike
hypersurfaces is necessary not only for the analysis of the dynamics along the
way, but also by the boundary conditions in an action principle. It is summed
∗
" plays the role of q.
Here, A
2.3.
The ADM Formalism
27
up in [8]: “Give the 3-geometries on the two faces of a sandwich of spacetime
and adjust the 4-geometry in between to extremize the action.”
Given the 3-geometry† of the “lower” hypersurface, gij (t, x, y, z)dxi dxj
gives the distance squared between two points on that hypersurface. Similarly for the “upper” hypersurface we have gij (t + dt, x, y, z). Then a formula
for the proper length is N (t, x, y, z)dt of the connector at the point (x, y, z),
where N (t, x, y, z) is the lapse function. This is properly defined as
N = −tα nα = (nα ∇α t)−1 ,
where tα is a vector field on the manifold satisfying tα ∇α t = 1, and nα is the
normal unit vector. The vector field tα can be interpreted as the flow in time
in spacetime.
Another thing we define is the shift vector N α by
N α = h α β tβ ,
where hαβ is the induced metric hαβ = gαβ + nα nβ . In this framework, N
measures the rate of flow of proper time τ with respect to coordinate time t
(i.e. dτ = N dt) and N α measures the amount of shift tangential to the 3geometry contained in the time flow vector field tα . The lapse function and the
shift vector are not considered to be dynamical, since they merely prescribe a
way to move “forward in time”.
We also need a formula for the place on the “upper” hypersurface
i
xupper (xm ) = xi − N i (t, x, y, z)dt where this connector is to be welded and
N i (t, x, y, z) is the so-called shift function. Now, by making use of the
Pythagorean theorem in its 4-dimensional form, we get
2
ds
=
3
proper distance
in base 3-geometry
42
−
3
proper time from
lower to upper 3-geometry
42
= gij (dxi + N i dt)(dxj + N j dt) − (N dt)2 .
This reasoning can be extended to more dimensions in a completely straightforward manner. Putting it all together the 4-metric should have the following
form:
3
4 3 !
4
"
g00 g0k
Ns N s − N 2 Nk
=
.
gi0 gik
Ni
gik
In order to lower the indices of the shift functions we used Ni = gim N m . The
inverse metric is
3
g 00 g 0m
g k0 g km
4
=
3
N m/N 2
−1/N 2 '
(
N k/N 2
g km − N k N m/N 2
4
.
Here, Latin letters indicate d − 1 geometry of a d-dimensional manifold. In this case
i, j = 1, 2, 3. Greek indices are used for the d-dimensional description; in our discussion, for
the 4-geometry.
†
28
Action Principle and Geometry
We can make the following connection. A timelike unit normal vector in
covariant 1-form will look like
nβ = (−N, 0, 0, 0).
By use of the inverse metric we can obtain the tangent vector
nα = (1/N , −N m/N ) ,
so that the typical “perpendicular” connector will have the components
(dt, −N m dt)
with proper length dτ = N dt.
So, we can write
nα =
1 α
(t − N α )
N
(2.14)
and express the inverse metric as
g αβ = hαβ − nα nβ = hαβ − N −2 (tα − N α )(tβ − N β ) .
For convenient we will choose the spatial metric hαβ , the lapse function N
and the covariant form of the shift vector Nα = hαβ N β instead of the inverse
metric g αβ as field variables. We see that these contain exactly the same
information. We can also write
√
√
−g = N h .
We now want to express the gravitational action in terms of (hαβ , N, Nα )
with their time and space derivatives as a first step in obtaining the Hamiltonian functional for general relativity. We will ignore boundary terms for the
sake of simplicity.
We start with the Hilbert action
ˆ
√
S=
−gLG
M
an express the Ricci scalar R as
R = 2(Gαβ nα nβ − Rαβ nα nβ ) ,
where for Gαβ we have
Gαβ nα nβ =
(
1 '(3)
R − Kαβ K αβ + K 2 ,
2
2.3.
The ADM Formalism
29
where Kαβ is the extrinsic curvature of the hypersurface and K = K α α . From
the definition of the Riemann tensor, we have
Rαβ nα nβ = Rαγβ γ nα nβ
= −nα (∇α ∇γ − ∇γ ∇α )nγ
= (∇α nα )(∇γ nγ ) − (∇γ nα )(∇α nγ )
−∇α (nα ∇γ nγ ) + ∇γ (nα ∇α nγ )
= K 2 − Kαγ K αγ − ∇α (nα ∇γ nγ ) + ∇γ (nα ∇α nγ ) .
The last two terms are divergences and will be discarded. Thus, we obtain
√
LG = N h((3) R + Kαβ K αβ − K 2 ) .
(2.15)
The extrinsic curvature is related to the “time derivative”, ḣαβ ≡ hα γ hβ δ £t hγδ
by
Kαβ =
=
=
=
1
1
£n hαβ = (nγ ∇γ hαβ + hαγ ∇β nγ + hγδ ∇α nγ )
2
2
1 −1
γ
N [N n ∇γ hαβ + hαγ ∇β (N nγ ) + hγβ ∇(N nγ )]
2
1 −1 γ δ
N hα hβ (£t hγδ − £N hγδ )
2
(
1 −1 '
N
ḣαβ − Dα Nβ − Dβ Nα ,
2
where Dα is the derivative associated with hαβ and we used (2.14). If we
substitute back to (2.15) we obtain the gravitational action in the desired
form given in [5].
The result however gives a LG that does not contain any time derivatives
of N or Nα . Thus, their conjugate momenta vanish identically. In analogy to
electromagnetism, we interpret this as the fact that N and Nα are not to be
viewed as dynamical variables. We define our Hamiltonian density by
HG = π αβ ḣαβ LG
%
&
1
= −h N R + N h
π παβ − π 2 + 2π αβ Dα Nβ
2
* %
&
.
'
(/
1 −1 2
1/2
(3)
−1 αβ
= h
N − R + h π παβ − h π − 2Nβ Dα h−1/2 π αβ
2
1/2
(3)
−1/2
αβ
+ 2Dα (h
−1/2
Nβ π
αβ
) ,
where π = π α α . The last term contributes as a boundary term and will be
dropped. Variation of HG with respect to N and Nα yields
1
−(3) R + h−1 π αβ παβ − h−1 π 2 = 0
2
Dα (h−1/2 π αβ ) = 0 .
+
(2.16a)
(2.16b)
30
Action Principle and Geometry
The dynamical equations (2.16a), (2.16b) obtained from HG are
δHG
ḣαβ = αβ
δπ
δHG
π̇ αβ = −
δhαβ
%
&
1
= 2h
N παβ − hαβ π + 2D(α Nβ)
2
%
&
1 αβ
1/2 (3) αβ
= −N h
R − Rh
2
%
&
1
1 2
−1/2 αβ
γδ
+ Nh
h
πγδ π − π
2
2
&
%
1
−1/2
αγ
β
αβ
−2N h
π πγ − ππ
2
−1/2
'
+h1/2 Dα Dβ N − hαβ Dγ Dγ N
'
(
(2.17a)
(
+h1/2 Dγ h−1/2 N γ π αβ − 2π γ(α Dγ N β) , (2.17b)
where again boundary terms have been dropped and (2.16b) was used. Equations (2.16) and (2.17) are equivalent to the vacuum Einstein equation Rµν =
0. This provides a constrained Hamiltonian formulation of Einstein’s equation.
However, there is still a gauge arbitrariness in our choice of configuration
field hαβ . This is due to diffeomorphisms. If φ is a diffeomorphism on the
hypersurface, then hαβ and φ∗ hαβ represent the same physical configuration.
What we should do is to take the configuration space of general relativity to
be the set of equivalence classes h̃αβ of Riemannian metrics on the hypersurface, where two metrics can be considered equivalent up to a diffeomorphism
transformation. So, for a vector field wα on the hypersurface the conjugate
momenta π αβ must satisfy
ˆ
'
( ˆ
π αβ δhαβ + D(α wβ) = π αβ δhαβ .
This implies that π αβ automatically satisfies
Dα (h−1/2 π αβ ) = 0
and therefore the constraint (2.16b) is eliminated by this choice of configuration space. The constraint (2.16a) though remains. It may be viewed as
the result of the gauge arbitrariness involved in the choice of how to “slice”
spacetime into space and time. It is analogous to the constraint which arises
when one “parametrizes” an original unconstrained theory in a fixed background spacetime. In other words, when one introduces a time function in
the Lagrangian, which defines the choice of hypersurfaces with respect to a
reference surface and treats this time function as a dynamical variable. In
such parametrized theories, the constraint (2.16a) is linear in the momentum
conjugate to the time function [9]. If this is the case, one can “deparametrize”
the theory by solving the constraint for this momentum. Unfortunately, in
2.3.
The ADM Formalism
Einstein’s equation, the constraint is quadratic in momentum and a similar
“deparametrization” appears impossible. It seems impossible to find a suitable
choice of configuration space in general relativity, such that only the “true”
dynamical degrees of freedom are present in its phase space. The constraint
(2.16a) is an intrinsic and unavoidable feature of the Hamiltonian formulation
of general relativity. This provides a serious obstacle when one tries to formulate a quantum field theory of gravitation following the canonical quantization
approach (for details see [9]).
31
3
AdS/CFT
Correspondence
The gauge-gravity refers to a fascinating equivalence between certain theories
with gravity and certain theories without gravity. For a bit over a decade
there has been intense research on a realization of such a duality, known as
the AdS/CFT correspondence, based on a seminal paper [10] by Maldacena,
which provides a duality between a theory with quantum gravity in d + 1
dimensions and a field theory in d dimensions. In particular, the claim is
that string theory in a Anti-de Sitter (AdS) background and a conformal filed
theory (CFT) are equivalent. In this chapter we will review the idea discussing
what motivated Maldacena to formulate the conjecture.
3.1
A first hint
The first hint of a relation between d dimensional conformal field theory in
Minkowski space is the fact that it has the same symmetry group as gravity
in d + 1 dimensional gravity in AdS space. Let us see how this comes about.
Conformal Invariance
In flat d dimensions (i.e. on R1,d-1 ), conformal transformations are defined by
xµ → x%µ (x) :
dx%µ dx%µ
=
[Ω(x)]−2 dxµ dxµ
It is noteworthy that what we mean by conformal invariance is not the same
as general coordinate invariance∗ since the metric is modified, from flat ds2 =
dx%µ dx%µ to “conformally flat” ds2 = [Ω(x)]−2 dxµ dxµ , yet we are studying flat
space field theories. In other words, conformal transformations are generalizations of the scale transformations that change the distance between the points
with the all-important property that all angles are preserved. Hence the name
conformal.
The conformal group be viewed as the set following transformations:
• Scale transformations: xµ → xµ + λxµ
∗
Although conformal transformations are a subclass of general coordinate transformations.
33
34
AdS/CFT Correspondence
• Translations: xµ → xµ + aµ
• Lorentz (rotations): xµ → xµ + ωµ ν xν with ωµν = −ωνµ
and the less familiar
• special conformal transformation: xµ → xµ + bµ x2 − 2xµ b · x.
So, the symmetry group has the following generators: Pµ for aµ and Mµν
for ωµν which together form the Poincaré group. The new generators are
Kµ for the special conformal transformations parametrized by bµ and the
dilatation generator D parametrized by λ. We can assemble the generators in
an antisymmetric (d + 2) × (d + 2) matrix
M̄M N
with
M̄µ,d+1 =


Mµν
M̄µ,d+1 M̄µ,d+2


=  −M̄ν,d+1
0
D

−M̄ν,d+2
−D
0
Kµ −Pµ
,
2
M̄µ,d+2 =
Kµ +Pµ
,
2
M̄d+1,d+2 = D .
The Lie algebra of M̄M N shows that the metric in the d + 2 direction is
negative, thus the symmetry group is SO(2, d). So conformal invariance in
(1, d − 1) dimensions (d > 2) corresponds to the symmetry group SO(2, d)† .
For completeness we write down the Lie algebra of the conformal group:
[Mµν , Pµ ] = i (gνρ Pµ − gµρ Pν )
[Mµν , Mρτ ] = i (gµτ Mνρ + gνρ Mµτ − gµρ Mντ − gντ Mµρ )
[Mµν , Kρ ] = i (gνρ Kµ − gµρ Kν )
[D, Pµ ] = iPµ
[D, Kµ ] = −iKµ
[Pµ , Kν ] = 2i (gµν D + Mµν )
with all others equal to zero.
Anti-de Sitter Space
AdS space is a space of Lorentzian signature but of constant negative curvature. Thus, it is the Lorentzian analogue of the so-called Lobachevski space‡ .
In d dimensions, de Sitter space is defined as a submanifold of Minkowski
Strictly speaking, SO(1, d+1) is the connected component of the conformal group which
includes the identity. The conformal group is an extension that also contains the inversion
x
I : x"µ = xµ2 ⇒ Ω(x) = x2 .
‡
Euclidean space with a constant negative curvature.
†
3.1.
A first hint
35
space in one higher dimension. Let us take Minkowski space R1,d with the
standard metric:
ds2 = −dx20 +
d−1
2
dx2i + dx2d+1 .
i=1
Then de Sitter space is the submanifold described by
2
R =
−x20
+
d−1
2
x2i + x2d+1
(3.1)
i=1
with some positive constant R. This is the Lorentzian version of the sphere,
and it is invariant under the group SO(1, d).
Similarly, in d dimensions, anti-de Sitter space is defined by a Lobachevskilike embedding in d + 1 dimensions
ds2 = −dx20 +
and
d−1
2
i=1
− R2 = −x20 +
dx2i − dx2d+1 .
d−1
2
i=1
x2i − x2d+1 .
(3.2)
It is therefore the Lorentzian version of a Lobachevski space. So, AdS5 can
be thought of as product of four-dimensional Minkowski space times an extra
radial coordinate. The metric on Minkowski space is however multiplied by
an exponential function of the radial coordinate. AdS space is therefore an
example of a warped space since in a suitable local coordinate system the
metric is
ds2 = r2 ηµν dxµ dxν +
dr2
.
r2
(3.3)
One can see that it is invariant under the group SO(2, d − 1) that rotates the
coordinates xµ = (x0 , xd+1 , x1 , . . . , xd−1 ) by x%µ = Λµ ν xν .
So AdS space in d + 1 dimensions has the same symmetry group as a
conformally invariant field theory in d dimensional Minkowski space!
Another useful choice of coordinates for the AdSd+1 are the so-called
Poincaré coordinates where metric is given by
ds2 =
R2
z2
'
(
(ηµν dxµ dxν )d+1 + dz 2 ,
z≥0
(3.4)
The boundary at spatial infinity (r → ∞) corresponds to z = 0 since z ∼ R2 /r
and the horizon at r = 0 corresponds to z = ∞.
36
AdS/CFT Correspondence
3.2
String Theory
In the previous sections we showed that conformally invariant field theories
enjoy invariance under the same symmetries as AdS space in one extra dimension. From this point of view, we could suspect that the same symmetries
suggest similar physics. We will now turn to string theory to argue that this
is indeed true.
In string theory, the fundamental objects are no longer point-particles.
Fundamental particles are understood as excitations of one-dimensional objects (either open or closed strings). It is a theory that consistently describes
a quantum theory of gravity. Unfortunately, despite its mathematical rigour
and robustness it seems impossible to check the theory experimentally with
present day technology.
The theoretical assumption of one-dimensional fundamental objects, allows
five different consistent descriptions in 10-dimensional spacetime, which lead
to five flavors of string theory. These are all considered equivalent and are
related through a web of dualities. In the reasoning that follows, we will start
with one flavor of string theory called type IIB string theory.
Dp-branes
When open strings are considered it is necessary to impose boundary conditions. There are two kinds of boundary conditions. Namely:
• Neumann boundary conditions.
In this case the component of the momentum normal to the boundary
of the world sheet vanishes. i.e.
;
∂Xµ ;;
= 0.
∂σ ;σ=0,π
If this holds ∀µ, these boundary conditions respect d-dimensional invariance under Poincaré transformations. Physically, Neumann boundary
conditions mean that no momentum is flowing through the ends of the
string.
• Dirichlet boundary conditions.
In this case the positions of the two string ends are fixed so that δX µ = 0
and
X µ |σ=0 = X0µ
and
X µ |σ=π = Xπµ
with X0µ and Xπµ constants and µ = 1, ..., d − p − 1. Neumann boundary conditions are imposed for the other p + 1 coordinates. Dirichlet
3.2.
String Theory
37
boundary conditions break Poincaré invariance. The constants X0µ and
Xπµ represent the positions of Dp-branes (D stands for Dirichlet). By
Dp-brane we mean a hypersurface (a p-brane) on which a fundamental
string can end. The presence of a Dp-brane breaks Poincaré invariance
unless it is spacetime filling (p = d − 1).
In 1995 Polchinski [11] proved that what we would strictly call D-branes are
exactly extremal p-branes (hence the hybrid name Dp-branes). By doing so,
he showed that the dynamical endpoints of open string correspond to extremal
solutions of supergravity, the low energy limit of string theory.
D-branes turn out to be dynamical objects that carry energy and therefore
curve space. The so-called black D3-brane (p = 3) supergravity solution for
N coincident D3-branes is:
!
ds2 = H −1/2 d,x2 + H 1/2 dr2 + r2 dΩ25
"
(3.5)
R4
, R = 4πgs N α%2
r4
Where R is the radius, α% is the Regge slope equal to the string length
squared ls2 . The first part of the metric describes the coordinates on the D3branes (d,x2 is the four-dimensional Minkowski metric on the brane). The
second part contains the coordinates perpendicular to the branes. The metric
S 5 is written dΩ25 , as usual. The horizon is at r = 0 and the near horizon
geometry is AdS5 × S 5 . This is the supergravity metric for the spacetime
where open strings propagate.
In the low energy limit the excitations near r = 0 and at spatial infinity
decouple from each other and we obtain one free supergravity theory and the
near horizon region the geometry of AdS5 × S 5 .
H(r) = 1 +
...
Fig. 3.1: Stack of D3-branes and open strings with endpoints attached to the hypersurfaces.
38
AdS/CFT Correspondence
Field theory of open strings
Now let us consider the case were open strings can have endpoints on these
D-branes. There will be i = 1, . . . , N possible states for each endpoint. Since
an open string has two endpoints, it has two so-called Chan-Paton indexes ij.
An open string state can be written as:
|p; a) =
N
<
i,j=1
λaij |p; ij)
The λaij matrices are called the Chan-Paton factors. Amplitudes obtained
when including Chan-Paton factors are invariant under local U (N ) transformations in spacetime. This is exactly what is required for Yang-Mills theories,
so it provides a basis for including the standard model in string theory. Considering coincident D-branes gives rise to massless gauge fields in the following
way:
• For N coincident Dp-branes, there are N 2 massless gauge fields§ .
• The open string massless states give an N = 4 vector supermultiplet in
(3 + 1) dimensions and the low effective Lagrangian is that of a N = 4,
d = 1 + 3, U(N ) Super Yang-Mills (SYM) theory on the world-volume
of N coincident D-branes [12] .
For the solution (3.5) we implicitly started off with type IIB string theory and
we considered a low energy limit where the theory on the D3-brane decouples
from the bulk. By the low energy limit we mean the following: we keep the
type IIB string coupling gs and the energies fixed by taking the equivalent
limit:
α% → 0,
U≡
r
α"
= fixed .
The N = 4 SYM on the D3-branes however, is not enough to give a full
description of the resulting physics. Two more things need to be considered.
1. There are closed strings living in the bulk. This gives supergravity coupled to the massive modes of the string. In the low energy limit only
supergravity survives.
2. Two open strings living on a D-brane may collide and form a closed
string. The closed string is no longer confined on the D-brane and may
move away as Hawking radiation.
§
A string gains mass from its tension. In the limit where the branes coincide the strings
become massless.
3.3.
The AdS/CFT Duality
39
This means that there should be a relation between the theory of open strings
living on the D-branes (N = 4 SYM) and the gravity theory of the bulk.
Qualitatively, the action of these strings would look something like:
S = Sbulk + Sbrane + Sinteractions
In the low energy limit α% → 0, the massive string modes drop out
√ and
we have Sbulk → SSUGRA and Sbrane → SN =4 SYM . Meanwhile Sint ∝ GN ∼
gs α%2 → 0. So we effectively get two decoupled systems: free gravity in the
bulk and N = 4, d = 1 + 3 SYM on the D3-branes.
3.3
The AdS/CFT Duality
Since we started from the same theory, namely type IIB string theory, and
using two descriptions we obtained–in the low energy limit–two decoupled
theories. In both cases one of the decoupled systems is supergravity in flat
space. It is therefore natural to assume that we can identify the second system
that appears in both descriptions.
The conjecture essentially states that since the starting point was a quantum theory that includes gravity the correspondence is valid beyond the supergravity approximation. In [10] the statement is: “Type IIB string theory
on AdS5 ×S 5 plus appropriate boundary conditions is dual to N = 4, d = 1+3
U(N ) Super Yang-Mills”. More specifically, the U(1) subgroup of U(N ) decouples as a free theory and does not participate in the duality. In particular, the
U(1) vector supermultiplet includes six scalars which are related the center
of mass of all the D-branes. From the AdS space point of view these zero
modes live at the boundary, and it looks like we might or might not decide to
include them in the AdS theory. Depending on this choice we could have a
correspondence to an SU(N ) or a U(N ) theory. We therefore have the choice
to interpreted U(1) as living on the boundary and the SU(N ) living in the
bulk, which is why the U(1) is not necessarily relevant. So the gauge group in
the duality can effectively be SU(N ) as a result of the large N limit¶ .
The next step in Maldacena’s approach was to identify the parameters of
the two theories and relate them. On the one side we have the Yang-Mills
coupling constant gY M and the number of colors N of the non-Abelian group
SU(N ). On the gravity side we have the type IIB solution in AdS5 × S 5 which
has the string coupling constant gs and the radius R. These parameters were
related by Maldacena in the following way
gs ≡ gY2 M
R4
a"2
≡ 4πgY2 M N ≡ 4πλ
Essentially we can write U(N ) ∼ SU(N ) × U(1). This effectively means that L →
LU(1) + LSU(N ) . But U(1) is Abelian and therefore the gauge fields commute; consequently,
the U(1) factor does not feel the strong gauge interactions .
¶
40
AdS/CFT Correspondence
where λ is the ’t Hooft coupling.
We can distinguish two limits where the duality becomes useful:
1. The large N limit
also known as the ’t Hooft limit, for which we have N → ∞ and gY2 M N =
λ / 1 and fixed. In this limit the string coupling (gs = λ/N ) tends to
zero. So it is possible to perform string calculations at tree level, the
classical limit of string theory.
2. The strong coupling limit
in this case we take λ → ∞ which makes the string tension very large.
This accounts for making all the massive modes extremely heavy. If we
keep the AdS radii fixed, we are dealing with the case were ls2 = α% → 0.
Strictly speaking one cannot send the dimensionful string length to zero.
What one means by this is to consider energy scales for which string
excitations may be neglected. If we consider the string length fixed and
still take λ large, the radius of S 5 tends to infinity. Then all curvatures
are “small” and quantum gravity corrections may be neglected: classical
supergravity is adequate.
Another property related to the strong coupling limit is supersymmetry. Quantum field theories at strong coupling are susceptible to severe instabilities. An
example of that is when particle and antiparticle pairs can appear spontaneously with their negative potential energy exceeding their positive rest and
kinetic energies. Supersymmetric theories however have a stabilizing property
because of the nature of the Hamiltonian& .
Because all this, it seems impossible to prove the conjecture. This would
require non-perturbative solutions of either the N = 4 SYM or the string
theory in the AdS5 × S 5 . But we do not have a good definition of nonperturbative type IIB string theory. Even at tree level we do not know how
to solve the theory completely. However, if the duality is true, this feature is
exactly what makes the conjecture so powerful. Stated differently, it is always
possible to choose the description where perturbative methods (and therefore
calculations) are possible. Moreover, it is possible and advantageous to view
the N = 4 SYM as the definition of non-perturbative type IIB string theory
on the AdS5 × S 5 background.
More intuitively, it is useful to bear in mind that the metric (3.3) at the
limit of the radial coordinate goes to infinity the exponential part blows up.
That is what we call the boundary of AdS space and there is where the dual
field theory lives. String theory excitations extend all the way to the boundary
and in this way one obtains a map from string theory states to states in the
field theory living on the boundary.
#
The Hamiltonian is the square of a Hermitian supercharge and therefore bounded from
below.
3.4.
The AdS/CFT Dictionary
41
Finally, it is worth mentioning, that the duality has not been proven wrong
either. It has been argued that the conjecture is so audacious that it should
be easy to disprove. Instead, qualitatively, we obtain exactly what we would
physically expect, for a wide variety of situations∗∗ . So, despite a lot of attention, the conjecture has resisted invalidation. On the contrary, compelling
evidence have accumulated.
3.4
The AdS/CFT Dictionary
In the original paper [10] Maldacena did not provide the exact correspondence
between elements of the bulk and the boundary theory. This was done by Gubser, Klebanov and Polyakov [14] and independently by Witten [15]. We start
by making some observations. A conformal field theory does not have particle
(massive) states or an S-matrix. If a mass was introduced, that would automatically infuse a scale and break conformal invariance. Moreover, the lack
of massive states, renders the S-matrix singular. The only physical observables, that is, well-defined and meaningful quantities, in a CFT are correlation
functions of gauge-invariant operators. What is necessary is an explicit prescription for relating such correlation functions to computable quantities in
the bulk theory. The operators are defined at a point. This corresponds to
perturbing the gauge theory in the ultraviolet. As explained in Chapter 4 this
amounts to considering the gauge theory at the boundary of the AdS space.
More concretely,
if we consider the corresponding operator O we can add
´ 4
the term d x φ0 (,x) O (,x) to the Lagrangian. It is natural to assume that
this will change the boundary condition of the dilaton at the boundary of
AdS to φ (,x, z)|z=0 = φ0 (,x). As argued in [14, 15] we can write for the string
theory full partition function:
= ´
e
d4 x φ0 (x)O(x)
>
CFT
.
/
= Zstring φ (,x, z)|z=0 = φ0 (,x) .
(3.6)
The left hand side is the generating function of correlation functions in the
field theory side. So, we can calculate correlation functions of O by taking
functional derivatives with respect to φ0 and then setting φ0 = 0. Each
differentiation brings down an insertion O, which “sends” a closed string state
into the bulk. Feynman diagrams can then be used to compute the interactions
of particles in the bulk. In the classical supergravity limit, the only diagrams
that contribute are the tree-level diagrams of the gravity theory (Fig. 3.2).
This is a rather general method for defining the correlation functions of
a field theory dual to a gravity theory in the bulk. In principle it applies to
any theory of gravity [15]. Each field propagating in AdS is in a one to one
correspondence with a single-trace operator in the field theory. Let us consider
∗∗
For example, the quark gluon plasma in RHIC physics is treated as an application of
the AdS/CFT correspondence without knowing the gravity dual of QCD[13].
42
AdS/CFT Correspondence
a scalar field φ of mass m in Euclidean AdS5 . Close to the boundary the wave
equation has two independent solutions which behave like z d−∆ and z ∆ with
d
∆= +
2
?
d2
+ R2 m2
4
determined by the equations of motion. In order to get consistent behavior
for the massive field the boundary condition for the partition function of (3.6)
should be changed to
φ (,x, +) = lim +d−∆ φ0 (,x) .
$→0
This leads to assigning a scaling dimension to φ0 of the form [length]∆−d so
that φ is indeed dimensionless. This implies that the associated operator O
has dimension ∆.
Fig. 3.2: Correlation functions can be calculated in terms of classical supergravity
Feynman diagrams. Here we see the leading contribution coming from a disconnected
diagram plus connected pieces involving interactions of the supergravity fields in the
bulk of AdS . At tree level, these diagrams and those related to them by crossing are
the only ones that contribute to the four-point function [16].
It is noteworthy that since the CFT always has a stress-energy tensor
Tµν as gauge invariant operator. This corresponds to the metric gµν in the
bulk. Therefore the AdS/CFT correspondence always involves a gravitational
theory for the bulk. Another bulk field is the dilaton which corresponds to
the Lagrangian of the CFT. This is because a small change in the gauge
coupling which is dual to the string coupling determined by the dilaton, adds
an operator proportional to the Lagrangian. Meanwhile, massless gauge fields
in the bulk correspond to global currents in the field theory.
A more general dictionary of various elements and their corresponding
analogs is given in Table 3.1. This is by no means complete but not because
of negligence. In fact, the known dictionary relating spacetime concepts in the
bulk and field theory concepts on the boundary is still being developed. So,
this is merely an example of some “entities” that have a proper corresponding
description on both sides of the duality.
3.4.
The AdS/CFT Dictionary
43
Corresponding elements
Gravity (bulk)
field φ
Gauge theory (boundary)
operator O
graviton gµν
Energy-momentum tensor Tµν
mass of the field
dimension of operator
gauge symmetry
global symmetry
gauge field
Chern-Simons term
Isometry
conserved current
anomaly
conformal symmetry
Table 3.1: Corresponding elements in AdS/CFT
4
Holographic
Renormalization
The holographic RG is based on the idea that the radial coordinate of a spacetime with asymptotically AdS geometry can be identified with the RG flow
parameter (i.e. energy scale) of the boundary field theory.
Let us consider a scale transformation of the gauge theory xµ → axµ . Scale
invariance implies that, if this is accompanied by a rescaling of the energy scale
E → E/a, this is a symmetry. Since xµ is identified with xµ in the bulk, this
scaling can be performed in the AdS metric (3.4). But when xµ → axµ , then
z → z/a for the scaling to be a symmetry of the metric. This leads to the
identification:
E ∼ 1/z ∼ r.
This shows that the radial coordinate in the bulk corresponds to the energy
scale of the dual gauge theory (this was realized in [10, 14, 15, 17]). By
dimensional analysis we get E = krls−2 , where k is a dimensionless constant.
One way to define it is by identifying the energy of a string stretched from
the horizon at r = 0 to a point with radial coordinate r with scale E. But
regardless of how one determines the constant, the important fact is that
E1 /E2 = z2 /z1 = r1 /r2 .
A reasonable question, given the identification, is where is the dual gauge
theory located. A theory on
without any degrees of freedom integrated out corresponds to E → ∞. In
this case the dual gauge theory is located at the boundary (r → ∞). When
high-momentum degrees of freedom are integrated out, it is translated inwards
toward the horizon.
This notion makes the correspondence more powerful. It can be generalized
to bulk theories that are only AdS asymptotically as r → ∞. On the other
hand, the dual gauge theory need only be conformal in the sense that it should
approach a conformal fixed point in ultraviolet.
Given this collection of ideas it should be apparent that talking about
renormalization in the context of the AdS/CFT correspondence is in fact
relevant. We are no longer looking at gauge theories were the β functions
vanish at any scale. As long as β → 0 in the ultraviolet regime, then a dual
description can become relevant.
The first systematic development of holographic renormalization for bulk
gravity coupled to scalar fields was given in [18]. This method involves the
45
46
Holographic Renormalization
cancellation of all cut-off related divergences from the bulk on-shell action by
the addition of counterterms on a cut-off boundary hypersurface and followed
by the removal of the cut-off. Later this method was summarized by Bianchi,
Freedmann and Skenderis [19].
Another approach was developed by de Boer, Verlinde and Verlinde (dBVV)
[20] (for reviews see [21, 22]). In their work the Hamilton-Jacobi theory is used
to separate terms in the bulk on-shell action, which can be written as local
functionals of the boundary data. The remaining, in principle non-local, part
was identified with the generating functional of a boundary field theory. This
approach is substantially different, since the boundary field theory lives on
the cut-off boundary and the generating functional contains logarithmic divergences. Notwithstanding, the dBVV method provides an amazingly simple
bulk description of the renormalization group flow in deformed CFTs and also
yields the correct gravitational anomalies. Additionally, the Hamilton-Jacobi
equation directly characterizes the classical action of bulk gravity without
solving the equations of motion.
Moreover, two apparent disadvantages in the original formulation of the
dBVV method were dealt with in [23]. More specifically, they attempted to
explain exactly how the ambiguities from the solutions of the local terms can
be removed and how logarithmic counterterms can be obtained. Martelli and
Mück also include U(1) gauge fields in their treatment which provides a useful
extension to the original formulation for scalar fields coupled to gravity.
4.1
Hamilton-Jacobi applied to gravity
We now discuss the Hamilton-Jacobi formalism as applied to gravity in its
canonical form. We essentially reformulate what was discussed in section 2.2
in a way relevant and useful to what follows. In the ADM formalism (section
2.2) the bulk metric can always be written as
ds2 = N 2 dr2 + gµν (x, r)(dxµ + N µ dr)(dxν + N ν dr) .
In this context the role of r will be similar to that of a time coordinate. Therefore, the Hamilton-Jacobi formalism will describe flows in the radial direction
instead of the typical flow in time. Locally, diffeomorphism invariance allows
us to choose N = 1 and N µ = 0, which is a common gauge to work in. The
Hamiltonian treatment involves the time-slicing formalism, which assumes
that the bulk spacetime manifold can be globally foliated into hypersurfaces
specified by a time coordinate (in our case r).
The Hamilton constraint emerges by imposing the equations of motion for
the lapse function N . Variation of the action with respect to N µ will give the
diffeomorphism constraint along a fixed time slice. The Hamilton constraint
will provide the Hamilton-Jacobi equation.
4.2.
The Holographic Renormalization Group
47
As toy model we will consider gravity minimally coupled to scalar fields φI ,
a potential V (φ) and a kinetic term 12 g µν ∂µ φI GIJ (φ)∂ν φJ . The Lagrangian
density will be of the form L = V (φ) + R + 12 g µν ∂µ φI GIJ (φ)∂ν φJ and the
variables of the theory will be gµν and the scalars φI at the cut-off (for the
classical trajectory). The corresponding canonical momenta will be πµν and
πI given by
1 δS
π µν = √
, πI
g δgµν
1 δS
=√
.
g δφI
(4.1)
The constraints in phase space are
1
1 δS
1
≡ H = π µν πµν −
π µ π ν + πI GIJ πJ + L = 0
√
g δN
d−1 µ ν 2
1 δS
≡ Hν = ∇µ πµν + πI ∇ν φI = 0 .
√
g δN ν
(4.2)
(4.3)
Given the expressions (4.1) we can recast (4.2) into the form:
√

3
1
1
δS
gL(φ, g) = √ 
g µν
g d−1
δgµν
42

δS δS
δS δS
− g µλ g νρ µν λρ − GIJ I J 
δg δg
δφ δφ
which gives (4.2) the desired form in terms of S[φ, g].
From a solution of the Hamilton-Jacobi equation, we can now compute
the radial derivatives of φI and gµν from the Hamilton equation of motion
q̇ = ∂H/∂p. Using (4.2) we obtain the coveted flow equations:
∂φI
∂r
∂gµν
∂r
4.2
= GIJ πJ
= 2πµν −
2
gµν πλλ .
d−1
(4.4)
The Holographic Renormalization Group
The divergences that arise in the supergravity action in the bulk of AdS need
to be dealt with. The divergent terms are local in nature which is exactly the
case in renormalizable field theories. According to [20] the action is non-local
at the scale of the cut-off , but at a much lower energy scale a part of it
can be represented as a local action. Following this reasoning the action is
decomposed in a local and a non-local part
S [φ, g] = Sloc [φ, g] + Γ [φ, g] .
(4.5)
48
Holographic Renormalization
Here, Γ represents the effective action of the gauge theory and contains all
higher derivative and non-local terms and Sloc includes all these local, divergent terms.
To make the discussion more concrete let us take
ˆ
&
%
1
√
Sloc [φ, g] =
g U (φ) + Φ(φ)R + ∂µ φI MIJ (φ)∂ µ φJ
2
where U , Φ and MIJ are local functions of the couplings. One needs to be
careful when defining S, since, in general it will be non-local. To this end,
a scaling procedure is required. What should not be done, is to simply see
how things rescale under rescalings of the metric. The reason is that the nonlocality of S will not make this work properly. The right thing to do is to
assign a degree +2 to gµν and a degree ∆I − 4 to φI . In order to motivate
this we should have the following in mind. In the Poincaré patch, the metric
of AdS5 × S 5 is described by
ds2 = dr2 + e2r/L ηµν dxµ dxν
which is invariant under
r → r + a,
!
ηµν → ηµν e−2a/L ,
"1/4
where L is the AdS5 radius L = gY2 M N
. A generic solution for the φ field
is
φ(r) ∼ αe(∆−4)r/L + βe−∆r/L + . . .
and ∆ satisfies ∆(∆ − 4) = m2 L2 . For large value of r0 , where the cut-off is,
we can perform the following substitutions
φ̂ ∼ er(∆−4) φ̂ , ĝ ∼ e−2a ĝ , r0 → r0 + a
which will leave the supergravity solution unchanged. Because of this transformation prescription we decide to assign the aforementioned degrees to gµν
and φI .
Now, Sloc is defined to be the local term that contains at least the complete
part of S that has a degree larger than zero. However, there is an ambiguity
in (4.5) since terms with zero and negative degree, in other words, finite local
terms may be shifted between Sloc and Γ. This is a manifestation of the
usual ambiguity of choosing a renormalization scheme, which was discussed in
section 1.5.
Fortunately though, the Hamilton-Jacobi theory takes care of that. As
we will see, it provides a way to choose an appropriate set of local terms in
Sloc and terms with positive degree are fixed unambiguously in this way. As
expected, this procedure makes sense only for marginal and relevant perturbations. Irrelevant perturbations may appear in terms of arbitrary positive
degree.
4.2.
The Holographic Renormalization Group
49
Now, the most critical step of this method is to split the on-shell action
based on power divergences:
'
S = S (0) + S (2) + . . . + S (2n)
(
loc
+ Γ,
(4.6)
where the power divergent terms are denoted by S (2k) , k = 0, . . . , n. Given
(4.1) the momentum π naturally splits into
π = π (0) + π (2) + . . . + π (2n) + π Γ
and similarly for π µν .
The key point of the dBVV method is to insert the expansion (4.6) into
the Hamilton constraint (4.2) and combine the contributions of the left hand
side that have the same scaling degree as the terms on the right hand side and
require them to cancel. This amounts to splitting the Hamilton constraint
into a derivative expansion
H = H(0) + H(2) + . . . + H(2n) + HΓ ,
where H(2k) denotes those terms in H that stem only from the counterterms
and contain a total of k inverse metrics. This is essentially a different way of
writing the Hamiltonian constraint in a form which makes the contributions
from the various counterterms explicit.
For this purpose we can define
ˆ
√
(0)
Sloc [φ, g] =
gU (φ)
(4.7)
ˆ
%
&
1
√
(2)
Sloc [φ, g] =
g Φ(φ)R + ∂µ φI MIJ (φ)∂ µ φJ
(4.8)
2
which is reasonable since
L(0) (φ, g) =
(2)
L
(φ, g) =
√
√
gV (φ)
%
1
g R + ∂µ φI GIJ (φ)∂ µ φJ
2
&
according to the power-counting prescription that we have.
The Hamilton-Jacobi equations then can be cast in the form:
{S, S} + L(0) + L(2) = 0 ,
where the brackets here denote:
0 %
1 1 µν δS
{S, S} = √
g
g 3
δg µν
&2
1
δS δS
1
δS δS
− g µλ g νρ µν λρ − GIJ I J .
δg δg
2
δφ δφ
50
Holographic Renormalization
By the scaling procedure explained, we have
@
(0)
(0)
Sloc , Sloc
@
(4)
A
@
(0)
(2)
2 Sloc , Sloc
@
(2)
(2)
2 Sloc , Γ + Sloc , Sloc
A
= L(0)
(4.9a)
A
= 0.
(4.9b)
A
= L(2)
The ambiguity of (4.5) is also reflected here in the sense that S (0) can also
contain terms of scaling weight≤ 0. Focusing on (4.9a) and collecting the
various terms we obtain
1
1
V = U 2 − GIJ ∂I U ∂J U .
3
2
(4.10)
This is the usual relation between scalar potential and superpotential of supergravity (for example [24]) which was recovered here by bosonic analysis.
We also obtain
φ̇I = GIJ ∂J U,
1
ġµν = − U gµν
3
(4.11)
from (4.4) where Poincaré invariant solutions
system alone
@ of the supergravity
A
(0)
(0)
(0)
(0)
are considered. In that case, S ≡ Sloc and Sloc , Sloc = L is the only nontrivial equation. Thus, we can recover all the information about the simple
flow directly from the Hamilton-Jacobi equation.
The following ansatz solves (4.11):
gµν = a2 ĝµν ,
with ĝµν independent of r and a satisfies
1
ȧ = − U (φ)a .
6
(4.12)
The parameter a determines the physical scale. We replace the r derivatives
in the flow equations by derivatives with respect to a using (4.12). We obtain
a
d I
φ = β I (φ)
dφ
with the following form for the β-functions
β I (φ) = −
6
GIJ (φ)∂J U (φ)
U (φ)
which are φI -dependent. Near the AdS boundary we have
β I (φ) ∼ (∆I − 4) φI + . . .
.
4.2.
The Holographic Renormalization Group
51
These β-functions do indeed play the role of β-functions in the renormalization
group equations.
IT should be noted that he solution of U to (4.10) is not unique. We can
identify U with the superpotential for supersymmetric flows, but this is not
a relation that generally holds. We can approach this issue perturbatively
around a critical point of V which will also be a critical point of U . We use a
basis where GIJ is δIJ and (4.10) can be rewritten as
1 2 1
U − (∂I U )2 = V .
3
2
(4.13)
By implicitly assuming that there exists a classical solution for V that can be
extended from the boundary to the interior, we get some restrictions on the
potential V . Apparently, it forces it to be of a supersymmetric form. We can
expand the 5-dimensional potential in powers of φI
1
V = 12 − m2I φI φI + gIJK φI φJ φK + . . .
2
where a possible linear terms in φI can be removed by the freedom to shift
the fields. We can attempt a similar expansion for U to obtain a solution to
(4.13) for the 4-dimensional potential,
1
U = 6 + λI φI φI + λIJK φI φJ φK ,
2
where the constant term has been chosen in order to match with that of V .
The β-functions are
β I = − (4 − ∆I ) φI − cIJK φJ φK
with
∆I = 4 − λI ,
cIJK = 3λIJK
(4.14)
the scaling dimensions and operator product coefficients of the operators OI
corresponding to the couplings φI . Inserting both expansions back to (4.13)
we obtain
λ2I − 4λI = m2I .
If we insert it back to (4.14) we obtain the standard relation (∆(∆ − 4) =
m2 L2 ) between the scaling dimensions ∆I of the 4-dimensional couplings and
the corresponding masses mI of the 5-dimensional fields. This relation implies
that the 5-dimensional potential must satisfy a unitarity bound m2I ≥ −4,
otherwise there will be no bounded solutions that extend to the asymptotic
boundary. This is of course the Breitenlohner-Freedman bound for stability in
AdS space. The existence of a perturbative solution implies that the masses
of the fields need to be consistent with an AdS solution.
52
Holographic Renormalization
Going to the next order we obtain
(λI + λJ + λK − 4λ) λIJK = −gIJK
which expresses the operator product coefficients cIJK in terms of the cubic
term in V .
An infinitesimal variation δU respects (4.13) provided that
4U δU − 6∂I U ∂I (δU ) = 0 or
'
(
4 + β I ∂I δU = 0 .
This means that the terms that have a total dimension 4 are not determined by
the Hamilton-Jacobi constraint. Interestingly, these are exactly those terms
that remain finite in the continuum limit. In other words, the Hamilton-Jacobi
relation apparently constraints only the divergent terms of the potential U but
not the finite part. These are the terms that survive as finite local terms in
the boundary effective action.
Moreover, there are discrete ambiguities in U since for example λI can be
either ∆I or 4 − ∆I . If the space is asymptotic AdS, the boundary conditions
in the infra-red limit will determine that the right solution is 4−∆I . In general
however, more general U can appear.
A
@
(0)
(2)
If we move to the next level of the action expansion and use 2 Sloc , Sloc =
L(2) we obtain the following equations
β K ∂K Φ = −2Φ +
−2MIJ +
6
GIJ
U
βI
6
U
= −12∂I ∂J Φ + β K ∂K MIJ − β K ∂I MKJ − β K ∂J MIK
= −6M IJ ∂J Φ .
In [23] a perturbative approach is also employed, but for only one field φ
but for d dimensions. The discussion then becomes a bit more involved in
settling the discrete ambiguities. In their notation and conventions:
V =−
d(d − 1) 1 2 2
1
1
+ m φ + υ3 φ3 + υ4 φ4 + O(φ5 ) ,
2
4L
2
3!
4!
where the constant part represents a negative cosmological constant. A similar expansion for U is used in order for each coefficient to be determined
recursively:
1
1
1
U = u0 + u1 φ + u2 φ2 + φ3 + φ4 + O(φ5 ) .
2
3!
4!
They find a breakdown for the procedure for λ = d/6, since
u3 =
2L
υ3 ,
6λ − d
4.2.
The Holographic Renormalization Group
53
which leaves the coefficient u3 undetermined. A more general breakdown will
occur for
(k − 2) d
λ=
2k
for k > 2. This will leave the uk coefficient undetermined and the boundary
integral of φk is finite in the limit where the cut-off goes to infinity. A similar
relation also occurs at higher levels.
The main idea so far is to find the power divergent terms by writing down
the most general set of covariant local counterterms up to the necessary level.
Then we solve for them by asking that they satisfy the Hamiltonian constraint
for arbitrary and independent values of boundary conditions of the fields. This
does not prove that the method will yield all power divergences. However,
there is no counter-example that suggests that the method will not.
Callan-Symanzik equation
The functional Γ contains all the information about the correlation functions
of the theory via the identity
1
δ
1
δ
... √
Γ [φ, g] ,
(OI1 (x1 ) . . . OIn (xn )) = √
g δφI1 (x1 )
g δφIn (xn )
(4.15)
for the case of many scalars. From this, the Callan-Symanzik equation can be
derived using (4.9b) which implies
%
1
δ
δ
√ 2g µν µν − β I (φ) I
g
δg
δφ
&
Γ [φ, g] = 4 − derivative terms.
(4.16)
What we need to do, in order to derive the Callan-Symanzik equations for
expectation values of local operators, we vary the last relation with respect
to the fields φI as we do in (4.15). After the variations, the fields are put to
their constant average value given by the couplings of the gauge theory.
If we take one extra step and assume the metric to be of the form gµν =
a2 ηµν , with a being xµ -dependent, the 4-derivatives will drop out. Therefore,
they will play no role, as long as operators at different points in space are
considered. Then the resulting expression is integrated over all space and the
functional derivatives are to be replaced by ordinary derivatives using
ˆ
ˆ
∂
δ
∂
µν δ
−2 g
=a
,
=
.
µν
I
δg
∂a
δφ
∂φI
This way, we can obtain the standard form of the Callan-Symanzik equations
from (4.16):
%
&
∂
a
+ β I ∂I (OI1 (x1 ) . . . OIn (xn ))
∂a
+
n
2
i=1
γIi Ji (OI1 (x1 ) . . . OJi (xi ) . . . OIn (xn )) = 0 , (4.17)
54
Holographic Renormalization
where γI J = ∇I β J represent the anomalous scaling dimensions of the operators OI .
In this procedure the Callan-Symanzik (4.17) still “carries” a finite cut-off.
We need to take the limit r → ∞. Thus, we write the metric and the couplings
as
R
gµν = +−2 gµν
and
φI = φI (φR , +)
where R denotes the renormalized metric and coupling which are kept fixed
in the limit + → 0. By integrating the RG-flow we obtain the relation between
the bare and the renormalized couplings
+
∂φI
= −β I (φ), φI = φIR , at + = 1 .
∂+
The renormalized effective action ΓR is defined by
.
ΓR [φR , gR ] = lim Γfinite φ(φR , +), +−2 gR
$→0
/
where Γfinite is obtained from Γ by subtracting the divergent part. Now the
action should again satisfy the Callan-Symanzik equation, but expressed in
terms of the renormalized couplings and metric
1
√
g
3
µν
2gR
δ
δ
I
µν − βR (φR )
δgR
δφIR
4
ΓR [φR , gR ] = local terms.
Beta-functions can be viewed as vector fields in the space of couplings as was
discussed in section 1.3, which means that
βI
δ
I δ
= βR
.
I
δφ
δφIR
If we also use the following definition of the renormalized operators
OIR = OJ
∂φJ
∂φIR
it is possible to obtain the Callan-Symanzik equations for all renormalized
n-point functions
%
&=
>
∂
I
a
+ βR ∂I OIR1 (x1 ) . . . OIRn (xn )
∂a
+
n
=
>
2
(R)
γIi Ji OIR1 (x1 ) . . . OJRi (xi ) . . . OIRn (xn ) = 0 .
i=1
4.2.
The Holographic Renormalization Group
Gauge Fields and Hamilton-Jacobi
In the application of the Hamilton-Jacobi theory to the case of non-relativistic
gauge/gravity duality, it is necessary to be able to treat gauge (vector) fields
as well as scalar fields as we shall see in the following chapter. To this end,
we present here the variation that includes vector fields in the original action.
We start by considering the following action:
ˆ
%
B
1
1
1
d+1
S = d x g̃ − R̃ + g µν ∂µ φ∂ν φ + V (φ) + K(φ)Fµν F µν
4
2
4
&
1 2
µ
+ M (φ)Aµ A
,
2
where we used tildes and Greek letters do denote d + 1 dimensions. The scalar
potential, V (φ), has a stable fixed point at φ = 0. The vector part of the
action also includes a massive gauge field. We employ the ADM formalism
once again, and we write the metric as
gµν =
3
ni ni + n2 nj
ni
gij
4
,
where we treat r as the zeroth component. The action now can be written as
ˆ
%
B
1
1
1
d+1
S = d x g̃ n − R + g µν ∂µ φ∂ν φ + V (φ) + K(φ)Fµν F µν
4
2
4
&
1 2
µ
+ M (φ)Aµ A .
2
We can gauge fix the following quantities
n = 1, ni = 0, Ar = 0 .
This way the corresponding equations of motion, enter as constraints into the
Hamilton formalism
δS
=H=0
δn
δS
= Hi = 0
δni
δS
=G
δAr
−
which translates into
1
1
1
1
1
H =
R + (∂r φ)2 + + Kg ij Fri Frj − g ij ∂i φ∂j φ − M2 Ai Ai
4
2
2
2
2
1
− KFij F ij − V (φ)
4
Hi = −∂r φ∂i φ − Kg jk Fij Frk
G = ∇i (KFri ) .
55
56
Holographic Renormalization
The gauge fixed action then reads
S=
ˆ
%
1
1
1
1
√
dd+1 x g − R + (∂r φ)2 + g ij ∂i φ∂j φ + V (φ) + Kg ij Fri Frj
4
2
2
2
&
1
+ M2 Ai Ai .
2
The congugate momenta are
1
δS
√
g δ(∂r gij )
1 δS
√
g δ(∂r φ)
1
δS
√
g δ(∂r Ai )
=
π ij
π =
Ei =
through which we can rewrite the constraints
1
4
1
1
1
πiπj + π2 +
Ei E i + R − g ij ∂i φ∂j φ − V
d−1 i j 2
2K
4
2
1 2
1
− M Ai Ai − KFij F ij = 0
2
4
j
= 2∇j πi − π∂i φ − Fij E j = 0
H = 4πji πij −
Hi
G = ∇i E i = 0 ,
where H coincides with the canonical Hamiltonian density, and G = 0 is the
Maxwell equation for the electric field in the vacuum.
The bulk theory is defined on a bulk spacetime with a boundary at r = ρ,
with ρ being the cut-off parameter. In the Hamilton-Jacobi formalism the
momenta of the theory are obtained from the on-shell action S as a functional of the prescribed boundary data, gij (x, ρ), φ (x, ρ) and Ai (x, ρ) from
the appropriate variations
π ij
=
π =
Ei =
1 δS
√
g δgij
1 δS
√
g δφ
1 δS
√
g δAi
with respect to the boundary data.
In order to discuss the equations obtained from the constraint H = 0 , it
is convenient to group the terms of the local part of the on-shell action into
4.2.
The Holographic Renormalization Group
57
different levels. The lowest possible counterterms for the vector sector are of
level two and read
ˆ
%
&
1
1
d √
(2)
ij
ij
Sυ = d x g
N (φ)g Ai Aj + P (φ)g ∂i φAj .
(4.18)
2
2
After computing the momenta that arise from the sum of (4.7), (4.8) and
(4.18) and inserting them into the Hamilton constraint it is evident that the
level zero equation is unchanged, the level two contributions can always be
solved by setting P (φ) = 0. The justification for this, is that we do not expect
counterterms linear in Ai . This way the gravity-scalar part of the analysis
remains the same.
The new equation to be solved is
Hυ(2)
=
3
U %N % d − 2
N2
M2
−
UN +
−
2
d−1
2K
2
4
Ai Ai = 0 .
We similarly expand U , N and M2 in powers of φ. For the flows that are of
interest to us in the last chapter, K(φ) = O(φ) and M2 (φ) = O(φ2 ). If we
use the solution we obtained for U we get
N0
%
N0
d−2
+
2K0
2L
&
= 0.
For asymptotically massless vector fields we should have N0 = 0 [23]. Moving
to a higher order in φ we get
N1 = 0
L
M2 .
λ−1 2
Clearly a breakdown occurs for λ = 1 and we have the remainder term
N2 =
1
Hrem = M22 φ2 Ai Ai + O(φ3 , A4 )
for λ = 1 .
2
Proceeding to the next levels increases the number of invariants dramatically,
making the complete analysis extremely tedious. On level four however, the
resulting equations can be solved by setting to zero the coefficients of all the
terms except for Fij F ij [23]. This leads to the equation
(4)
HF
=
%
&
d−4
1
G0 − K0 Fij F ij = 0 ,
4L
4
where G0 is the constant part of the coefficient of the Fij F ij counterterm.
The equation breaks down for d = 4 leading to a logarithmic divergence of Γ
and a contribution to the conformal anomaly [23]. Thus, up to level four, the
anomaly contributions from the vector sector are
A=
-
− 21 LM22 φ2 Ai Ai
1
ij
4 LK0 Fij F
(λ = 1)
(d = 4) .
5
Non-relativistic
Holography
Despite the auspicious realization of a specific theory for a holographic description of a physical system, many things remain to be achieved. Most
importantly, and regardless of many attempts, the promise of managing to
describe a strongly correlated system that is experimentally tangible is yet to
be fulfilled. Notwithstanding the contribution to a qualitative understanding
of real-time dynamics and transport properties of the quark-gluon plasma in
QCD, no holographic dual matching the precise microscopic details of any
such system has emerged.
In search of a possible dual of a strongly correlated system that can be
engineered in the laboratory, attention was drawn to non-relativistic systems.
This field provides a plethora of strongly correlated systems which can be experimentally studied in detail. Moreover, some of these are of extraordinary
technological interest. The focus would of course be on non-relativistic conformal field theories arising from these systems that would potentially have
a gravity dual. So, the ambition is to develop a holographic approach to the
non-relativistic theories that would describe condensed matter systems.
In particular, the question might be stated as which field theories with
Galilean scaling symmetry ([25] and references therein and thereto) have a
holographic dual. Along the lines of generalizing the Poincaré algebra to the
conformal algebra, one could extend the Galilean algebra to the so called
Schrödinger algebra.
An example of a system (at least conjectured) to realize this Schrödinger
symmetry is fermions at unitarity. To achieve the desired scale invariance one
can fine tune the interactions (with an external magnetic field for example) to
obtain a massless two-body boundstate. This effectively leads to an infinite
scattering length, therefore to a strongly correlated system.
As we described in Chapter 3 the “original” AdS/CFT [10, 14, 15] correspondence maps relativistic conformal field theories holographically to gravitational dynamics in a higher dimensional asymptotically AdS spacetime. So, by
pushing the rules of gauge-gravity duality beyond the case considered by Maldacena, the idea is to get a gravity dual of a non-relativistic field theory. The
gravity dual of non-relativistic CFTs was proposed almost simultaneously by
Son [26] on one hand and independently by Balasubramanian and McGreevy
[27] on the other. The goal is to demand respect of the Schrödinger algebra
which is obtained from the relativistic conformal algebra by reducing along
a light-cone. The procedure resembles light-cone quantization, where for a
59
60
Non-relativistic Holography
fixed light-cone momentum only a Galilean subgroup of the Lorentz group is
manifest.
Contrary to the approach of [10], the proposals of Balasubramanian, McGreevy and Son have a different starting point. The fact that the conformal
group in the context of field theories coincides with the isometries of AdS
spacetime in one extra dimension was known before the statement of the
AdS/CFT conjecture. However, Maldacena, even though he did not prove
the duality, constructed it in a very consistent and convincing manner directly through careful considerations in a string theory context. The setting
in these proposals is somewhat different. Here, the gauge/gravity duality is
a given and the starting point. Not bothering with a string theory construction, the writers accept the idea that non-relativistic CFTs do indeed have a
holographic gravitational dual. They directly provide the gravitational duals
focusing only on symmetry considerations. Embedding in string theory followed in [28, 29, 30]. Following the structure of Chapter 3 we first discuss
the so-called Schrödinger group and then the corresponding geometry. We
then present the bulk theory and briefly outline an example of a string theory
embedding.
5.1
Schrödinger Group
The Schrödinger group essentially is the Galilean group which includes the
spatial translations Pi , rotations Mij , Galilean boosts Ki , and time translations H, with the addition of a dilatation operator D and a particle number
(or conserved rest mass) operator N . Particle numbers can be conserved and
this is related to the fact that we cannot have pairs of particles-antiparticles
creation in a non-relativistic theory. This does not exclude the presence of antiparticles in a non-relativistic theory as long as it is invariant under t → −t.
But this will not be the case in what we will consider here.
Scale invariance can be realized in a number of ways. We will take advantage of the freedom of the relative scale dimension of time and space, called
the “dynamical critical exponent” and which we denote as z. Assuming spatial
isotropy, we can in general have the scaling action
(t, ,x) → (λz t, λ,x)
λ ∈ R,
(5.1)
where z can in principle take any positive value but for the Schrödinger case,
as we will see, it is z = 2. The dynamical critical exponent was first introduced
as an anisotropic space and time scaling of the renormalization group [31] and
we will discuss its meaning in the following.
5.1.
Schrödinger Group
61
The relevant algebra is given for completeness:
[Mij , N ] = [Mij , D] = 0, [Mij , Pk ] = i (δik Pj − δjk Pi ) ,
[Mij , Kk ] = i (δik Kj − δjk Ki ) ,
[Mij , Mkl ] = i (δik Mjk − δjk Mil + δil Mkj − δjl Mki )
[Pi , Pj ] = [Ki , Kj ] = 0, [Ki , Pj ] = iδij N, [D, Pi ] = iPi ,
[D, Ki ] = (1 − z) iKi , [H, N ] = [H, Pi ] = [H, Mij ] = 0, [H, Ki ] = −iPi ,
[D, H] = izH, [D, N ] = i (2 − z) N .
Where i, j = 1, . . . , d label space dimensions. The last commutator indicates
that the case of z = 2 is rather special. At this value of z, the dilatations
commute with the number operator. Therefore, both D and N can be diagonalized, so representations of the Schrödinger algebra can be in general
labeled by the scaling dimension ∆ and one more number 2. For fermions at
unitarity this is precisely the fermion (particle) number. This commutator is
the mathematical expression of the fact that mass is dimensionless at z = 2
(maintaining ! = 1). This is why the free Schrödinger equation can be scale
invariant with this particular time and space scaling. Moreover, the z = 2
case allows for an extra “special conformal” generator, C, to be added to the
algebra with non-trivial commutators
[D, C] = −2iC ,
[H, C] = −iD .
It is however noteworthy that apart from the free Schrödinger theory there
are known examples of interacting theories which respect the Schrödinger
symmetry at a quantum level and are known as non-relativistic conformal
field theories [25].
The Dynamical Critical Exponent
Before we discuss the holographic construction of the non-relativistic conformal field theories it is worth mentioning a few interesting facts about z and
its physical consequences. We will start by providing a somewhat crude but
rather intuitive way of looking at z. By making the traditional identifications
for energy we see that
∂
∼ λ−z ,
(5.2)
E∼
∂t
based on the rescaling described by (5.1). Similarly for the momentum one
would have
∂
p∼
∼ λ−1 .
(5.3)
∂x
Meanwhile, if we were to examine a free Hamiltonian we could insist that it
will depend on the momentum to an arbitrary power #. That is
H ∼ p# .
62
Non-relativistic Holography
We immediately see from (5.2), (5.3) that this arbitrary number # has to
be equal to z. So, at least at a superficial level, the dynamical critical exponent
can be viewed as the power of the momentum that appears in the Hamiltonian
of a system. This reasoning is in full agreement with the Schrödinger equation
(z = 2). In the same spirit, it also agrees
with the relativistic equation that
B
2
relates energy and momentum, E = p + m2 for z = 1. We can keep this in
mind when we encounter the gravity dual of a NRCFT for a general z. We
should expect that the metric will turn into that of an AdS spacetime when
we set z = 1.
Perhaps the most important aspect of z is that it has the following physical
consequence: it determines the critical dimension of interactions. To address
this, we will closely follow [32]. In this discussion, time and all frequencies
that will appear will be Euclidean, because the real time description of actions
which are non-analytic in frequencies is quite subtle.
Suppose we have a free field theory of the form
ˆ {Λ,Λ} d−1
(
d k dω '
2/d
2
r
+
k
+
|ω|
|Φ (ω, k)|2
S=
d
(2π)
where Λ is the UV cutoff for both frequencies and momenta and Φ is an
N component vector. Firstly, we integrate out modes with momenta and
energies between some lower cutoff Λ% and the original cutoff Λ. Due to the
anisotropic rescaling between time and space, the trick is to lower the energy
and momentum cutoffs by different amounts:
Λ%k = e−l Λ, Λ%ω = e−zl Λ, for some l > 0 .
The action then becomes
ˆ {Λ" ,Λ"ω } d−1
(
k
d k dω '
2/d
2
S=
r
+
k
+
|ω|
|Φ (ω, k)|2 + const .
(2π)d
The second step according to the renormalization procedure is to rescale the
momenta, energies and the field Φ in order to restore the action to its original
form with a rescaled value of the “coupling” r. If we let
!
"
k % = el k, ω % = ezl ω, Φ% ω % , k % = e−(z+d+1)l/2 Φ (ω, k) ,
then the action becomes
ˆ {Λ" ,Λ"ω } d−1
k
; % ;2/d ( ; % ! % % ";2
d k dω ' 2l
%2
;ω ;
;Φ ω , k ; + const .
S=
re
+
k
+
(2π)d
(5.4)
This shows that the theory can be renormalized to lower energies and momenta
by the rescalings (5.4). If we add a quartic interaction
ˆ 1/Λ
' (2
Sint. =
dd−1 x dτ u Φ2 ,
5.2.
Holographic Construction
63
it would be of interest to know whether this interaction becomes stronger
or weaker as we flow to lower energies. From (5.4), noting that the Fourier
transform implies Φ% (τ % , x% ) = e(z+d−3)l/2 Φ (τ, x), we have
u% = e(5−z−d)l u .
So, the coupling u becomes weaker at low energies (irrelevant) if
d > dc = 5 − z .
For z = 1 we recover the result that the critical spacetime dimension of
relativistic Φ4 theory is d = 4. Interestingly though, for z > 1 we observe
that the critical dimension is lowered. This fact was first noted in [31] and
implies that quantum critical points are increasingly tractable by perturbative
methods. In [32], one can find an incomplete but useful list of systems that
are described by different values of z with brief explanations.
5.2
Holographic Construction
The goal would be to be able to study strongly coupled Galilean-invariant
conformal field theories using the gauge/gravity duality. In order to do this
we need to realize the Schrödinger algebra geometrically. However, we now
have two symmetry generators which may be diagonalized simultaneously and
whose eigenvalues label nonequivalent representations (in the usual AdS case,
there is only the scaling dimension). This leads to pushing the AdS/CFT
“rules” slightly: since it is impossible to arrange for the whole algebra of a
d dimensional Schrödinger invariant field theory to act as the isometries of
a d + 1 dimensional spacetime we must consider a candidate dual in d + 2
dimensions! Such geometries where explicitly constructed in [26, 27]. The
metric is (following the notation of [27]):
2
2
ds = L
3
dt2 d,x2 + 2dξdt dr2
+ 2
− 2z +
r
r2
r
4
(5.5)
which is invariant under the anisotropic scale transformation
'
(
(r, ,x, t, ξ) → λr, λ,x, λz t, λ2−z ξ .
Here, ,x is a d-vector and the generators of the Schrödinger algebra are
geometrically given by
Pi = −i∂i ,
'
H = −i∂t ,
(
Ki = −i −t∂i + xi ∂ξ ,
N = −i∂ξ .
'
(
Mij = −i xi ∂j − xj ∂i ,
'
(
D = −i zt∂t + xi ∂i + (2 − z) ξ∂ξ + r∂r , (5.6)
64
Non-relativistic Holography
The last identification is perhaps the least obvious, but rather important. The
particle number is given by the momentum in the ξ direction. It is common in
the AdS/CFT correspondence for global symmetries in field theory to appear
as extra dimensions in the gravitational dual. The fact that the ξ direction
is null ( |∂ξ |2 = 0) and that the N generator arises in the commutator of
two spacetime symmetries is rather unusual. We also note once more that
for the special case where z = 2 the dilatations commute with the number
operator. It is worth mentioning that the metric (5.5) is not invariant under
time reversal, but it is under the combined operation
t → −t,
ξ → −ξ .
Given the interpretation of the ξ-momentum as rest mass we can interpret
this combination CT as charge conjugation and time reversal.
What is important here is that in systems of physical interest the number
operator (i.e. the spectrum of masses) is quantized. This tells us that ξ in the
bulk description must be periodic:
ξ ∼ ξ + 2πLξ .
(5.7)
This identification introduces a mass scale. Unless z = 2 we can see from
(5.6) that dilatations will not preserve the length Lξ and hence are no longer
isometries of the background. It is therefore impossible to have a scale invariant Galilean theory with a nontrivial discrete mass/particle number spectrum
for z %= 2.
If we compactify ξ, boost invariance remains unbroken. precisely
because
/
the ξ direction is null; this follows from the commutator N̂ , K̂ = 0 in the
Schrödinger algebra [29]. Having a circle becoming very small in our geometry
may render the calculations unreliable. This is because the identification of
the null direction leads to a zero proper length, the supergravity regime cannot
be trusted. Fortunately, when considering a sector with large non-zero lightcone momentum (along our null direction ξ) one finds regions in the geometry
where the circle has a non-zero size, so that the calculations can be trusted
[30]. This potential conical singularity for r → ∞ which suggests unreliability
of the metric in the IR, turns out to be unphysical. More specifically, the
singularity goes away as soon as we turn on finite temperature. This is,
physically, a perfectly reasonable situation: we will always have some finite T
in a realistic cold-atom system, and thus an IR regulator. In [30] it is argued
that by considering in a finite number density resolves potential problems and
it is obviously a physically sensible thing to do. In [29] it is mentioned that
even in the T → 0 limit the dynamics will resolve the singularity in a way
already familiar from the study of null orbifolds of flat space [33, 34, 35, 36].
5.2.
Holographic Construction
65
Causal Structure and Global Coordinates
Another important characteristic, is that the metric (5.5) has the causal structure that naturally reproduces the Galilean light-cone of the field theory. The
causal structure of a non-relativistic theory is intrinsically degenerate. The
metric (5.5) shares this degeneracy and therefore is consistent with the boundary Galilean invariance. This a crucial ingredient in the gauge/gravity duality.
A bulk theory with a well behaved causal structure cannot be holographically
dual to a non-relativistic theory or we would end up having inconsistencies
when calculating correlation functions. In our case this is ensured by the sign
of gtt , as all points at some fixed t = t0 share the same causal future and past
[28, 32].
However, it should be made clear that the spacetime in question is causal
in the sense that it doesn’t have closed timelike curves. Moreover, if the sign
of gtt is reversed while keeping the orientation of ξ fixed, the space can become
unstable to modes with large particle number [37]. Also, by reversing the sign
we lose non-relativistic causality. This is due to the fact that the dtdξ term
grows at the same rate as d,x when going towards the boundary r → 0. In
order for the lightcones to flatten the gtt term must be negative. For z %= 1, 2
a different sign can lead to geodesic incompleteness at the boundary and the
so-called pp singularities [37]. In [38] the “Schrödinger” metrics (for z = 2)
are shown to have a global, geodesically complete coordinate system and that
spacetimes with z > 2 admit no global timelike Killing vector fields. This
means that the global metric will necessarily be time-dependent.
Starting from the fact that the z = 2 algebra has the central element
P− = ∂ξ , we are looking for new coordinates
'
, V
(t, r, ,x, ξ) → T, R, X,
(
in which H + C and P− are simultaneously diagonal,
H + C = ∂T ,
P− = ∂V .
This is accomplished by the transformation
,
X
R
, ,x =
,
cosT
cosT
'
(
1
, tanT .
ξ=V +
R2 + X
2
t = tanT,
r=
This leads to the following form of the metric
ds2
0
3
,
1
X
= −
+
1
+
R4
R2
= −
41
dT 2 +
(
1 '
2
,
−2dT
dV
+
dR
+
d
X
R2
'
(
/
dT 2
1 .
2
, 2 dT 2 + dR2 + dX
,2 .
+
−2dT
dV
−
R
+
X
R2
R2
66
Non-relativistic Holography
A few comments are in order. First of all, it is geodesically complete. What is
most fortunate about the above transformation is that it is considerably more
simple than the AdS counterpart. For example, in the AdS case, instead of
the relation t = tan T used here, the Poincaré to global time transformation
is
2t
tan T =
.
1 + r2 + ,y − t2
A relevant and remarkable feature of the global metric is that the difference
between the Poincaré metric is only in a single term: the coefficient of dT 2 .
Again, comparing between the two, the AdS metric is substantially different
when expressed in Poincaré or global coordinates.
5.3
The Bulk Action
According to [27] the metric (5.5) is sourced by the following stress tensor
Tab = −Λgab − Eδa0 δb0 g00 .
It consists of a negative! cosmological
constant Λ and a pressureless “dust” of
"
constant density E = 2z 2 + z − 3 L−2 for d = 3. A negative cosmological
constant is a basic ingredient and as we shall see, so is a massive vector field.
Without further delay, we give the action that can be used to engineer the
metric (5.5):
S=
ˆ
%
'
(&
√
1
1
dd+3 x −g − F 2 + |DΦ|2 − V |Φ|2
,
4
2
where Da Φ ≡ (∂a + ieAa ) Φ , with a Mexican-hat potential
'
(
'
√
%
&
V |Φ|2 = g |Φ|2 − υ 2
(2
+ Λ.
This produces the correct dust stress tensor for arbitrary g as long as the
gauge field mass is m2A = e2 υ 2 = z(z+d)
.
L2
Similarly, Son [26] uses the same ingredients to generate the solution (5.5).
He suggests the action (in his notation):
S=
ˆ
d
d+2
x dz
1
1
m2
−g
R − Λ − Hµν H µν −
Cµ C µ ,
2
4
2
where Hµν = ∂µ Cν − ∂ν Cµ and of course Cµ is the gauge field. The desired
metric∗ together with
C− = 1
∗
#
$
In Son’s notation: ds = z −2 −2z −2 (dx+ )2 + −2dx+ dx− + dxi dxi + dz 2 .
5.4.
String Theory Embedding
67
and the following choices
1
Λ = − (d + 1) (d + 2) , m2 = 2 (d + 2)
2
is indeed a solution to the coupled Einstein and Proca equations.
Turning back to [27] we can observe that due to the non-stationary form
of the metric, the stress tensor for an electric field in the radial direction Frt
has only a 00 component.
Therefore, we consider a nonzero current j ξ of the form† j ξ = ρ0 rα . Then
we calculate (starting from Maxwell’s equation in the bulk):
(
'√
1
= jb ⇒
gF ab
√ ∂a
g
At =
ρ0
rα−2 .
(α − 2) (α − d − 2)
Thus, the ansatz At ∝ rz solves the equations of motion, provided that
1
Λ = − (d + 1) (d + 2) ,
2
5.4
m2 = z (z + d) .
String Theory Embedding
As already mentioned, the string theory embedding was achieved by [28, 30,
29]. We will present the procedure followed by [29] and mention similarities
with the procedure followed by [28]. It is somewhat beyond the scope of this
work to thoroughly discuss this issue, so we will simply mention some basic
aspects of it.
In the work of Herzog, Rangamani and Ross [28] and independently by
Adams, Balasubramanian and McGreevy [29] the same tool is employed to
achieve a string theory embedding. Namely, the Null Melvin Twist (Appendix
C), which is a solution generating technique for IIB supergravity.
Null Melvin Twist
Interestingly, some solutions with the desired general characteristics have already appeared using the Null Melvin Twist [39, 40]. For T = 0 one of the
appearing solutions yield a Schrödinger spacetime for z = 3. These theories,
which have some part of the symmetry group broken, are described as dual to
“dipole theories” which are also discussed [30]. The algorithm of the NMT is
given here and details can be found in Appendix C:
1. Choose a translationally invariant direction (e.g. y) and boost along it
by γ
†
Because of the term dtdξ in the metric, this choice will produve the desired result.
68
Non-relativistic Holography
2. T-dualize along y
3. Re-diagonalize isometry generators by shifting dφ → dφ + αdy
4. T-dualize along y again
5. Boost back along y by −γ.
We follow [29] and then discuss the [28] case which is similar. The starting
point will be the extremal D3-brane, which is a IIB supergravity solution with
the following metric:
ds2 =
where h2 = 1 +
F (5) =
R4
ρ4
(
'
(
1'
−dτ 2 + d,x2 + h dρ2 + ρ2 ds2S 5
h
is the D3 harmonic function, and self dual flux form
1
dτ ∧ dy ∧ dx1 ∧ dx2 ∧ dr + Ω5 dθ ∧ dφ ∧ dψ ∧ dµ ∧ dχ ,
r5
with Ω5 = 1/8 cos θ cos µ sin3 µ. We take dy to lie along the worldvolume and dφ
along S 5 and choose–without loss of generality–coordinates such that y = x3 .
A convenient choice for dφ is given by the Hopf fibration S 1 → S 5 → P2 (see
Appendix C), with the metric
ds2S 5 = ds2P2 + (dχ + A)2
with χ being a local coordinate on the Hopf fiber and A is the 1-form potential
for the Kähler form on P2 so that JP2 = dA. We take dφ = dχ. Both dy and
dφ act freely.
The result is (ignoring ds2P2 for now)
'
(
'
(
'
(/
1.
−dτ 2 1+β 2 ρ2 + dy 2 1−β 2 ρ2 + 2dτ dy β 2 ρ2 + hρ2 (dχ+A)2
h
B = 2βρ2 (dχ + A) ∧ (dτ + dy)
ds2 =
Φ =Φ
0,
with β = αc (see Appendix C). Nothing happend to the five-form, since Tdualizing takes dΩ5 , the top form on the sphere, to dy ∧ dΩ5 , so that the twist
dφ → dφ + βdy acts trivially. So,
F5 = (1 + 6) Ω5 dθ ∧ dφ ∧ dµ ∧ dχ .
To make the Schrödinger nature of the solution transparent we take a few
more steps. We change the coordinates according to
y+τ
t= √
2
y−τ
ξ= √
2
5.4.
String Theory Embedding
69
so that the background becomes
'
(
ds2 = h−1 β 2 ρ2 dt2 + 2dtdξ + hρ2 (dχ + A)2
B = 2βρ2 (dχ + A) ∧ dt
Φ =Φ
0.
Adding back everything we obtain
'
(
'
ds2 = h−1 −β 2 ρ2 dt2 + 2dtdξ + d,x2 + h dρ2 + ρ2 ds2S 5
B = 2βρ2 (dχ + A) ∧ dt
Φ =Φ
(
0.
As a final step, we take the near-horizon limit h → R2/ρ2 and switching to the
global radial coordinate r = R2/ρ, which makes h = r2/R2 the solution becomes
3
4
R2 2∆2 2
ds =
dt + 2dtdξ dr2 + R2 ds2S 5
r2
r2
√ ∆R2
B = 2 2 2 (dχ + A) ∧ dt
r
Φ =Φ 0 ,
2
with ∆ = βR2 . After compactifying on S 5 , we recover the desired Schrödinger
geometry with d = 2 and z = 2.
In [28] essentially they perform the same procedure and obtain effectively
the same result in a slightly different notation and in changed coordinates.
We mention their result here for completeness (in their notation)
'
(
ds2 = r2 −2dudυ − r2 du2 + d,x2 +
F(5) = 2 (1 + 6) dψ ∧ J ∧ J
dr2
+ (dψ + A)2 + dΣ24
r2
B(2) = r2 du ∧ (dψ + A)
(5.8)
in light-cone coordinates
u = β(t + y) ,
υ=
1
(t − y) ,
2β
(5.9)
where now, β = 12 αeγ with γ being the same as before.
However, they also consider non-extremal D3-branes and repeat the NMT.
So, the starting point is the planar Schwarzschild-AdS black hole times S 5 with
a F(5) flux
'
ds2 = r2 −f (r)dt2 + dy 2 + d,x2
(
1
+ 2
r
3
4
dr2
+ r2 dΩ25 ,
f (r)
70
Non-relativistic Holography
where again the S 5 will be written as a fibration over CP2 . Now the solution
generating machine leads to the string frame metric
ds2str
eφ
F(5)
= r
2
0
1
β 2 r2 f (r)
f (r) 2
dy 2
dr2
−
(dt + dy)2 −
dt +
+ d,x2 + 2 (5.10)
k(r)
k(r)
k(r)
r f (r)
(dψ + A)2
+
+ dΣ24
k(r)
1
= B
k(r)
= dC(4) = 2(1 + 6)dψ ∧ J ∧ J
B(2) =
r2 β
(f (r)dt + dy) ∧ (dψ + A) ,
k(r)
with
f (r) = 1 −
4
r+
,
r4
k(r) = 1 + β 2 r2 (1 − f (r)) = 1 +
4
β 2 r+
,
r2
(5.11)
with the obvious notation r+ for the horizon of the solution. The parameter
β appearing in this case is an independent physical parameter. The lack of
extremality has broken the boost symmetry. Therefore, we cannot boost along
the ty plane and set this parameter equal to unity, as we could normally do
in the extremal case.
5.5
The 5d effective Lagrangian
The five dimensional Lagrangian is developed also in [28]. The starting point
is the Lagrangian of Son [26] and how that is related to the ten dimensional
IIB theory. The effective Lagrangian is repeated here
S=
ˆ
d
d+2
%
&
1
1
x dr −g R − 2Λ − Fµν F µν − m2 Aµ Aµ .
4
2
√
(5.12)
If one starts from IIB supergravity and Kaluza-Klein (Appendix D for a discussion of KK compactifications) reduce the solution (5.8) on an undeformed
S 5 the procedure is straightforward and the result is the desired metric in five
dimensions. The two-form however has different behavior compared to the
case where a CP2 fibration is used, because it depends on the coordinates of
S 5 . Such a mode of the two-form produces a massive vector transforming in
the 15 of SO(6).
5.5.
The 5d effective Lagrangian
71
If a KK reduction is performed on (5.10)we obtain
.
ds2E = r2 k(r)− /3 −β 2 r2 f (r) (dt + dy)2 − f (r)dt2 + dy 2 + k(r)d,x2
2
1
+k(r) /3
2
dr2
r2 f (r)
−2/3
= r k(r)
1/3
+k(r)
3
0%
4
β 2 r+
1 − f (r)
2
2
−
r
f
(r)
du
+
dυ 2 − (1 + f (r)) dudυ
4β 2
r4
dr2
r d,x + 2
r f (r)
2
&
/
2
4
1
(5.13)
where the light-cone coordinates (5.9) were used in the second line, and with
massive gauge field and scalar
A =
=
eφ =
r2 β
(f (r)dt + dy)
k(r)
3
4
β 2 r4
r2
1 + f (r)
du − 4+ dυ ,
k(r)
2
r
1
B
,
k(r)
for the same f (r) and k(r) in (5.11). In these light-cone coordinates the solution asymptotically approaches the extremal solution (5.5), with β however
remaining a physical parameter, in the sense that the full metric depends on
β.
The metric (5.13) is a solution to
ˆ
%
&
1
4
1 −8φ/3
5 √
µ
µν
µ
S(5) =
d x −g R − ∂µ φ∂ φ − e
Fµν F − 4Aµ A − V (φ) ,
16πG5
3
4
with scalar potential
V (φ) = 4e2φ/3 (e2φ − 4) .
The black hole geometry is not a solution to (5.12). The scalar that appears is because of the non-vanishing dilaton of the black hole geometry and
melvinization causes the fibration over CP2 to be squashed. The squashing
can be intuitively ascribed to the distortion of the asymptotics of spacetime
as a feature of the NMT.
Although the writers of [28] determine this action and use it in their work,
they add it with a disclaimer. They introduce it, even though they have no
argument that the five-dimensional action describes a consistent truncation of
the full ten-dimensional theory. The mention that some modes that transform
non-trivially under SO(6) which are turned on in their ansatz may couple to
other Kaluza-Klein harmonics which they neglect. In favor of the ansatz
however, they mention that, by construction the black hole solution uplifts to
IIB supergravity.
6
Hamilton-Jacobi and
non-relativistic
holography
In the previous chapters we laid the ground so that we can finally attempt
to apply the holographic renormalization procedure, based on the HamiltonJacobi theory, to the non-relativistic version of gauge/gravity duality. Despite the extensive literature on renormalization group flows in AdS/CFT,
this framework has not yet been implemented to the non-relativistic case.
We shall make some formal developments and in the process, exclude some
possibilities.
6.1
Non-relativistic case
For the original case, in the context of AdS/CFT, the holographic renormalization procedure using the Hamilton-Jacobi theory was outlined in Chapter
4. In Chapter 5 we discussed some aspects of the more recent developments of
the gauge/gravity dualities that include non-relativistic theories. We saw in
section 5.3 that the basic ingredients of the bulk action in the newly proposed
duality–in both formulations [26, 27]–include a negative cosmological constant
and a massive gauge field.
So, as a first step, we would directly apply the method outlined in Chapter
4. In order to do so, we should first notice that the idea we used to split the
on-shell action as in (4.6) now becomes a bit more subtle. This is because
of different scaling for space and time. Thus, the scaling procedure we used
throughout Chapter 4 changes and depends on the dynamical critical exponent
z. In particular, what makes this entire procedure interesting, is the fact that
because of the form of the Schrödinger metric (z = 2), all the time componets
of the equations stemming from the Hamilton-Jacobi equation should “jump”
to the next level equation.
It is interesting to consider what we could expect to obtain from this
procedure. In [30] there is a discussion on the proposals of [26, 27] for nonrelativistic holography bearing in mind that a discrete light cone quantization
(DLCQ) of a field theory gives a non-relativistic system. Based on the known
example of performing a DLCQ on the M5 brane theory [41], it is discussed
that the DLCQ of a relativistic conformal theory with a gravity dual, is suggestive of performing the DLCQ of AdS space. It is also noted that subtleties
73
74
Hamilton-Jacobi and non-relativistic holography
occur when performing a DLCQ of gravitational backgrounds and the gravity
approximation cannot be naively applied.
In the present case, it would be interesting if we could retrieve some structure resembling the DLCQ limit of a theory as we approach the boundary.
We will see that it is not so easy to organize things in such a way as to make
this structure transparent.
6.2
Application
In order to apply the method we massage the original metric to bring it into
a convenient form. Starting from
ds2 = L2
3
dt2 2dtdξ + d,x2 + dr2
− 2z +
r
r2
4
we make the following transformation
dr2
dr
= dr̃2 ⇒
= dr̃ ⇒ r̃ = ln r ⇒ r = er̃ .
2
r
r
The substitution in (5.5) is then straightforward
'
ds2 = dr̃2 − e−2zr̃ dt2 + e−2r̃ 2dξdt + d,x2
(
but now the boundary is at r̃ → −∞ and we implicitly take L = 1. From now
on we will write r̃ as r. Now, the metric effectively has the desired form of
ds2 = dr2 + gij (x, r)dxi dxj where i, j = 1, 2 . . . , d + 2. It is also in the desired
gauge in the sense that the lapse and shift functions have the values N = 1
√
√
d+2
and N i = 0. We also have g = (e−r )
as opposed to g̃ = r−(d+3) for the
original metric (5.5).
Before moving on we will rewrite the Hamilton constraint for an action of
the general form which we derived in section 4.2:
S=
ˆ
d
d+1
x
%
1
1
1
g̃ n − R + g µν ∂µ φ∂ν φ + V (φ) + K(φ)Fµν F µν
4
2
4
&
1 2
µ
+ M (φ)Aµ A .
2
B
We temporarily assume a metric of the form
ds2 = N 2 dr2 + gij (x, r)dxi dxj
and the inverse metric is
g µν =
3
1/N 2 0
0
g ij
4
.
6.2.
Application
75
For the action we have chosen the variation yields the Hamilton constraint
1
1
1
1
1
H = R + (∂r φ)2 + M 2 A2r + K(φ)g ij Fri Frj − g ij ∂i φ∂j φ
4
2
2
2
2
1 2
1
i
− M Ai A − K(φ)Fij F ij − V (φ)
2
4
However, in our case, the inverse metric of (5.5) is




g µν = 
1 0
0
0
2r
0 0 e
0
0 e2r −1
0
0 0
0 e2r 1d×d



.

This, unfortunately, renders the procedure problematic from the start, because
according to [27] in order to get the desired stress tensor we need to have an
electric field in the r direction. Therefore, we consider a nonzero current j ξ
of the form j ξ = ρ0 rα . This yields (starting from Maxwell’s equation in the
bulk and using the original form of the metric):
'√
(
1
= jb ⇒
gF ab
√ ∂a
g
At =
ρ0
rα−2 .
(α − 2) (α − d − 2)
The important thing here is that the gauge field has a time component only.
Given the inverse metric, we see that all the remaining Maxwell related terms
in the variation of the action with respect to the lapse function vanish. Moreover, the expected interesting feature where the tt terms would appear in the
next higher level of the Hamilton-Jacobi equation is gone.
If we choose to ignore this, we can continue and see how the treatment of
the scalar potential looks like. Our potential has the form:
V (φ) =
'
(
gυ 2 + Λ − 2gυ 2 φ2 + gφ4 .
c ≡ gυ 2 + Λ.
We assume that U (φ) to be:
1
1
U (φ) = λ + λ1 φ2 + λ2 φ4 .
2
4
Using the equation for the superpotential, as discussed in Chapter 4
d
1
U 2 − (∂φ U )2 = V (φ) ,
d−1
2
(6.1)
76
Hamilton-Jacobi and non-relativistic holography
we have explicitly
3
4
λ2
λλ2 4 λ1 λ2 6
d
1
λ2 + 1 φ4 + λ22 φ8 + λλ1 φ2 +
φ +
φ −
d−1
4
16
2
4
'
(
λ21 φ2 + λ22 φ6 − 2λ1 λ2 φ4 = c − 2gυ 2 φ2 + gφ4 .
The next step is to determine each coefficient order by order. So,
dλ2
=c⇒
d−1
?
λ=±
%
d−1
c,
d
&
d
λ − λ1 λ1 φ2 = −2gυ 2 φ2 ⇒
d−1
λ1 =
dλ
d−1
±
C'
dλ
d−1
(2
dλ
2 d−1
− 8gυ 2
and
g =
λ2 =
λ21 λλ2
+
+ 2λ1 λ2 ⇒
4
2
2
g − λ1/4
.
λ/2 + 2λ1
In search of a different metric that would have the same desired characteristic, namely Schrödinger invariance, one could look for version with non-zero
temperature and/or non-zero chemical potential. An apparently relevant case
is the derived metric (??) of [28], which we rewrite here for convenience:
.
ds2E = r2 k(r)− /3 −β 2 r2 f (r) (dt + dy)2 − f (r)dt2 + dy 2 + k(r)d,x2
2
1
+k(r) /3
dr2
r2 f (r)
/
with
4
4
r+
β 2 r+
2 2
,
k(r)
=
1
+
β
r
(1
−
f
(r))
=
1
+
r4
r2
and with massive vector and scalar
f (r) = 1 −
A=
r2 β
(f (r)dt + dy) ,
k(r)
According to [28] this is a solution of
5 − d effective action:
ˆ
%
√
1
4
S=
d5 x −g R − ∂µ φ∂ µ φ −
16πG5
3
eφ = B
1
.
k(r)
the equations of motion from the
&
1 −8φ/3
e
Fµν F µν − 4Aµ Aµ − V (φ)
4
6.2.
Application
77
with a scalar potential
2φ/3
V (φ) = 4e
'
(
e2φ − 4 .
Before even discussing how to split the action based on power divergences
(i.e. looking at the metric asymptotics) we can simply look at the zeroth order
and treat the scalar potential like we did previously. Ideally, we would like an
analytic solution of 6.1. In principle, this equation does not necessarily have
an analytic solution. We can however, try the obvious ansatz
U (φ) = aecφ − bedφ .
We will now use the 5-dimensional version of the potential equation, as found
in [21]
1
1
V (φ) = U 2 − U %2
3
2
giving
%
&
%
&
%
&
1 1 2 2 2cφ 2 2dφ 1 1 2
cd 1 (c+d)φ
8φ
2φ
− c a e +b e
− d +2ab
−
e
= 4e /3 −16e /3 .
3 2
3 2
2
3
The obvious problem–the last term in the left hand side– is taken care of for
cd = 2/3. Inserting that back we get
%
&
1 1 2 2 2cφ
4
− c a e + be /3cφ
3 2
3
1 1
−
3 2
%
2
3c
&2 4
8φ/3
= 4e
2φ/3
− 16e
from where it is obvious that not both powers of the exponential can be correct
for a single choice of c. We need to use Taylor expansion. The potential has
the profile shown in Fig. 6.1
The Taylor expansion is
V (φ) = −12 +
32 2 320 3 224 4
φ +
φ +
φ + O(φ5 ).
3
27
27
Following the same steps as we did for the original action we assume a potential∗
1
1
1
U (φ) = λ + λ1 φ2 + λ2 φ3 + λ3 φ4 .
2
3!
4!
This results in an imaginary constant term for the potential U (φ). The interpretation of this is not so clear. We therefore abort this procedure and turn
to a more general way to determine the leading term.
∗
although stopping at φ3 also maintains a local minimum around φ = 0
78
Hamilton-Jacobi and non-relativistic holography
Fig. 6.1: Scalar potential for 5 − d effective action
6.3
Domain wall solutions
The general way to determine the relation between the V (φ) and U (φ) is inspired by supergravity but does not require supersymmetry, and it is outlined
in [42]. The motivation in [42] is phenomenological and they present a solution generating technique involving branes. In our case, we have no reason
to include branes. In fact, it would make it more complicated to remain consistent with the required symmetries. The fact that breaking the dilatation
symmetry results in a local action that consists only of the potential U (φ) is
something that simplifies the analysis to some extent.
On the other hand, we must reconsider the form of the metric trying to
respect as many of the symmetries of the group, but not dilatation invariance.
This results in assuming a metric of the following form:
'
ds2 = dr2 − e2b(r) dt2 + e2a(r) 2dξdt + d,x2
(
where b(r) and a(r) are some functions of r that have to be determined. We
further assume a general action of gravity, scalars and a Maxwell term. We
use the conventions of [42] for the metric (by adding the factor of two before
the functions a(r) and b(r)) and in the action, so that we can compare the
results. So, the action looks like:
ˆ
*
+
D
1
1
1
S=
d4 x dr | det gµν | − R + (∂φ)2 − V (φ) − F 2 ,
4
2
4
M
6.3.
Domain wall solutions
79
where M is the whole of spacetime. Rotational invariance allows for only
two components of the gauge field to be non-zero. This makes it pointless to
include a mass term for the gauge field for this class of metrics. It is also the
reason our fields and metric only have an r dependence. We can also gauge
away one component and we choose to retain the time component of the gauge
field which will be a function of the radial coordinate, f (r), only.
We can obtain the Ricci tensor:
'
(
i,j(=t
'
(
Rij = −e2a(r) a%% + a%2 ,
Rrr = −4 a%% + a%2 ,
'
(
Rtt = e2b(r) 2a% + b%% + 2b%2 ,
where we use primes to denote d/dr, and Rij includes the off diagonal entries
Rtξ . The equations of motion now read:
∂V (φ)
∂φ
2
= − φ%2
3
1
1
= − V (φ) + φ%2 ,
3
6
4a% φ% + φ%% =
a%%
a%2
(6.2)
which are identical to the equations in [42]. In addition, we have one more
equation for the tt component,
4a%% + 8a%2 + 2V (φ) + φ% = b%% + 2b%2 −
f2
2π
and Maxwell’s equations yield
'√
(
! !
"
"
1
gF ab = eb(r)−2a(r) f 2a% + b% + f % .
√ ∂a
g
So, we are left with a set of non-linear equations. The purpose of this
method is to reduce the system (6.2) to three decoupled first order ordinary
differential equations two of which are separable. This method becomes nontrivial when more than just one scalar are considered. One of the differential
equations has φ as the independent variable. For several scalars it would
become a difficult partial differential equation.
The idea is to assume that the potential V (φ) has the special form [42]:
1
V (φ) =
8
%
∂U (φ)
∂φ
&2
1
− U (φ)2
3
and verify that a solution to
φ% =
1 ∂U
,
2 ∂φ
1
a% = − U (φ)
3
80
Hamilton-Jacobi and non-relativistic holography
is also a solution to (6.2), which is true. For our purpose we need to generalize
the above approach to d dimensions and then proceed to solve the remaining
equations so as to define b(r) and f (r). This way we can relate the characteristics of the desired class of metrics to the potential U and finally deduce the
leading term in the local action that is related with power divergences.
7
Discussion
The extension of gauge/gravity duality in order to describe non-relativistic
systems is a promising endeavour. However, a lot remains to be done. There
are several subtleties involved with various aspects of non-relativistic holography that remain unclear. Perhaps the most intriguing aspect of this new
correspondence is the one extra dimension of the bulk description (in addition
to the usual radial direction). The null ξ direction which seems to be associated to conserved rest mass results in a variety of interesting features of the
theory.
In the present thesis we discussed the holographic renormalization procedure, based on the Hamilton-Jacobi theory, starting from the basic idea of
ordinary renormalization. The connection of the radial direction and the energy scale of the boundary theory in “traditional” AdS/CFT correspondence
is a key element in approaching the problem of renormalizing the theory. Using tools from general relativity applied to AdS/CFT, we presented a method
of holographic renormalization.
We discussed some aspects of the recently conjectured non-relativistic holographic duality and then proceeded to bring the machinery of holographic
renormalization to the non-relativistic framework. The anisotropic scaling of
time and spatial directions makes the synthesis of the non-relativistic holographic renormalization procedure fascinating in its own right. The suspicion
was that it would be possible to regain some structure resembling the discrete
light cone quantization of a field theory in the appropriate limit.
There are several issues that need to be resolved. The main problem in
the analysis is that the null direction since it renders the existing approach insufficient in maintaining information of the t components of various elements.
Although, several deformations of the metric were attempted, no promising
results emerged. Finally, a generalized description was adopted in order to
determine the leading term in the renormalization procedure. This approach
remains incomplete, but it appears adequate to provide some structure with
the desired properties. However, it remains an open problem for further research.
81
Acknowledgements
I would like to express my gratitude to my advisor, Jan de Boer for his help
and guidance throughout this last year. I would also like to thank Kyriakos
Papadodimas for his support and endless discussions on physics and beyond.
The following people have had the most pleasent and positive effect in
my life the past two years and I am happy to mention them here: P.M.
Adamopoulou, I. Angeletopoulos, J.S. Caux, A. Geromitsos, P. Koutroumpis,
D. Lykakis, P. McCullough, M. Panfil, D. Palioselitis, B.M. de Regt, A. Stergiou, M. Tsiakiris.
Finally, I would like to thank my family for their love and support.
83
Appendices
A
Dimensional
Regularization
Here we outline the idea of dimensional regularization. In a nutshell, what
we do is to compute a Feynman diagram as an analytic function of arbitrary
dimensionality, d. For sufficiently small d, any loop-momentum integral will
converge. The final expression for any observable quantity should have a welldefined limit as d → 4.
So for a loop momentum integral, we generalize its expression for d dimensions
ˆ
ˆ
ˆ
dd p
1
dΩd
pd−1
=
·
dp
.
(2π)d (p2 + m2 )2
(2π)d
(p2 − m2 )2
The area of a d-dimensional unit sphere is
ˆ
dΩd =
2π d/2
,
Γ(d/2)
where Γ is the gamma function.
We push this idea further to consider non-integer dimensionality, and write
the dimensions as d = 4 − +. So, we obtain
lim
$→0+
ˆ
dp 2π (4−$)/2
p3−$
'
(
.
(2π)4−$ Γ 4−$ (p2 + m2 )2
2
85
B
Anti-de Sitter space
The maximally symmetric d-dimensional spacetime with negative curvature
R
(κ = d(d−1)
< 0) is known as anti-de Sitter (AdS) space. It is the vacuum
solution of Einstein’s field equation with an attractive (negative) cosmological
constant Λ:
1
Rµν − R =
2
R =
Rµν
=
1
Λgµν
2
d
Λ
2−d
Λ
gµν .
2−d
So, in these spacetimes the Ricci tensor is proportional to the metric tensor
(Einstein spacetimes). Maximal symmetry in addition, suggests that
Rµνρσ =
R
(gνσ gµρ − gνρ gµσ ) .
d(d − 1)
Of course, in different coordinate systems the intrinsic properties must
remain the same. Calculations however, may be substantially simplified with
the right choice of coordinates. Moreover, certain characteristics can become
more transparent.
In order to describe AdS space in global coordinates we must follow a series
of transformations of the coordinates of (3.2). These are:
x0 = R cosh ρ cos τ , xd+1 = R cosh ρ sin τ , xi = R sinh ρΩi
where i = 1, . . . , d, ρ ≥ 0, 0 ≤ τ ≤ 2π, Ωi are the coordinates on S d−1 and
<
satisfy di=1 Ω2i = 1 and R is the AdS radius. Then
'
ds2 = R2 − cosh2 ρdτ 2 + dρ2 + sinh2 ρdΩ2i
(
where dΩ2i is the metric of the S d−1 . Near ρ 4 0 the metric becomes
'
(
ds2ρ→0 = R2 −dτ 2 + dρ2 + dΩ2i .
87
88
Anti-de Sitter space
Fig. B.1: AdS1+1 embedded in three dimensions
Another convenient choice of coordinates is the so-called Poincaré coordinates. This coordinate system can be introduced by first defining the light
cone coordinates:
u ≡
υ ≡
(x0 − xd )
,
R2
(x0 + xd )
.
R2
Redefining the other coordinates as
xi
Xi ≡
Ru
xd+1
t≡
Ru
(spacelike)
(timelike)
and so (3.1) takes the form
with ,x =
<d
2
i=1 xi .
R4 uυ + R2 u2 (t2 − ,x2 ) = R2
From this we find
1
(1 + u2 (R2 + ,x2 − t2 )
2u
1
=
(1 + u2 (−R2 + ,x2 − t2 )
2u
= RuX i
i = 1, . . . , d − 1
x0 =
xd
xi
xd+1 = Rut .
Anti-de Sitter space
89
It is now useful to change to the coordinate z ≡ u1 . In this way the Poincaré
coordinates z, ,x, t are defined as
x0 =
xd =
xi =
xd+1 =
1 2
(z + R2 + ,x2 − t2 )
2z
1 2
(z − R2 + ,x2 − t2 )
2z
RX i
i = 1, . . . , d − 1
z
Rt
.
z
In terms of these coordinates the AdS metric takes the form
ds2 =
R2
(dz 2 + d,x2 − dt2 ) .
z2
The coordinate z behaves as a radial coordinate and divides the AdS space
in two regions. The first chart is the region z > 0 and corresponds to the
one half of the hyperboloid. In global coordinates this region can be obtained
by imposing the condition sinh ρΩi < 1. The other half of the hyperboloid
x0 < xd corresponds to z < 0, or in global coordinates sinh ρΩi > 1. As we
see in Fig. B.2 the hyperplane x0 = xd cuts the entire AdS space.
Fig. B.2: AdS hyperboloid intersected by the hyperplane x0 = xd−1 .
C
Null Melvin Twist
Here we present the Null Melvin Twist and some mathematical notions relevant to Chapter 5.4
C.1
Complex Projective Space
Complex projective space, CPn sometimes denoted Pn , is a compact manifold
with n complex
dimensions.
It can be constructed by taking Cn+1 /{0}, that
! 1 2
"
is a set of z , z , . . . , z n+1 %= (0, 0. . . . , 0) and making the identifications
'
(
'
(
z 1 , z 2 , . . . , z n+1 ∼ λz 1 , λz 2 , . . . , λz n+1 ,
for any nonzero complex λ. These are homogeneous coordinates in the traditional sense of projective geometry. Thus, lines in Cn+1 correspond to points
in CPn . One may also regard CPn as a quotient of the unit 2n + 1 sphere in
Cn+1 under the action of U(1):
CPn = S 2n+1 /U(1) .
This is because every line in Cn+1 intersects the unit sphere in a circle. By
first projecting to the unit sphere and then identifying under the natural action
of U(1) one obtains Cn+1 . For n = 1 this construction yields the classical Hopf
bundle.
C.2
Hopf Fibration
As mentioned in Chapter 5.4 convenient choice is to realize S 5 as a Hopf
fibration over P2 .
The round metric on Pn and S 2n+1 can be expressed in terms of invariant
1-forms of SU(N ). In particular, for SU(3) we have
σ1 =
σ2 =
σ3 =
1
(dθ cos ψ + dφ sin θ sin ψ)
2
1
(dθ sin ψ − dφ cos ψ sin θ)
2
1
(dψ + dφ cos θ) .
2
In terms of the 1-forms, the metrics on P2 and S 5 can be written as
'
(
ds2P2
= dµ2 + sin2 µ σ12 + σ22 + σ32 cos2 µ
ds2S 5
= ds2P2 + dχ + σ3 sin2 µ
'
91
(
92
Null Melvin Twist
where χ is the local coordinate on the Hopf fiber and
A = sin2 µσ3 = x1/2 sin2 µ (dψ + dφ cos θ) is the 1-form potential for the Kähler form on P2 (dχ + A is the vertical 1-form along the Hopf fibration).
C.3
Buscher Rules and Conventions
%
=
gyy
Φ% = Φ −
C.4
1
% =
, gay
gyy
Bay
gyy
1
% =
ln gyy , Bay
2
gay
gyy
gay gyb + Bay Byb
gyy
gay Byb + Bay gyb
= Bab −
.
gyy
%
, gab
= gab −
%
, Bab
Algorithm Steps
We directly apply the Null Melvin Twist on the case of interest following [29],
we start off by using the black D3-brane solution
'
ds2 = h−1 −dτ 2 f + dy 2 + d,x2
(
0
1
'
(
dρ2
+h
+ ρ2 ds2P2 + (dφ + A)2 ,
f
where h2 = 1 + R4/ρ4 is the D3 harmonic function and f = 1 + g = 1 − ρ4H/ρ4 is
the emblackening factor. Since in the following steps the terms d,x2 , dρ2 and
ds2P2 will remain intact so we will not carry them around, but they will be
reintroduced when the calculation is complete. Thus, the metric we will work
on looks like
'
(
ds2 = h−1 −dτ 2 f + dy 2 + hρ2 (dφ + A)2 .
• We now boost by γ, so that τ → cτ − sy with c = cosh γ and c2 − s2 = 1,
.
'
(
'
(
/
ds2 = h−1 −dτ 2 1 + gc2 + dy 2 1 − gs2 + 2dτ dy gcs
+ hρ2 (dχ + A)2 .
• Then, we T-dualize along the dy isometry using the Buscher rules
*
f
1
+ h ρ2 (dχ + A)2 + dy 2
h (1 − gs2 )
1 − gs2
%
&
−gcs
B = 2dy ∧ dτ
1 − gs2
ds2 = −dτ 2
1
Φ =Φ 0 − ln
2
3
1 − gs2
h
4
.
+
C.4.
Algorithm Steps
93
• Shift dχ → dχ + αdy, where α has dimension [L]−1 ,
0
!
2 2
2
f
2
2
2 1 + ρ α 1 − gs
ds = −dτ
+
h
ρ
(dχ
+
A)
+
dy
h (1 − gs2 )
1 − gs2
2
2
2
2
+ 2dy αρ (dχ + A)
• T-dualize back along dy
2
ds
0
−dτ 2
g 2 c2 s2
f
−
h (1 − gs2 )
1 + ρ2 α2 (1 − gs2 )
=
*
+
"2
1
1
0
dydτ
gcs
dy 2
1 − gs2
+2
+
h
1 + ρ2 α2 (1 − gs2 )
h 1 + ρ2 α2 (1 − gs2 )
.
'
+hρ2 (dχ + A)2 1 + ρ2 α2 1 − gs2
(/−1
.
1
.
'
( /
αρ2
2
(dχ
+
A)
∧
gcs
dτ
+
1
−
gs
dy
1 + ρ2 α2 (1 − gs2 )
B =
1
Φ =Φ 0 − ln
2
3
1 − gs2
h
4
.
• Boost back by −γ and take the limit α → 0 with αc = β held fixed
(β ∼ [L]−1 ). We also reintroduce the terms that were suppressed in the
following steps since they remained unchanged
ds2 =
'
(
'
(
/
1 .
−dτ 2 1 + β 2 ρ2 f + dy 2 1 − β 2 ρ2 f + 2dτ dy β 2 ρ2 f
hK
0
1
2
2
dρ
ρ
2
+ ρ2 ds2P2 +
+h−1 d,x + h
(dχ + A)
f
K
2βρ2
(dχ + A) ∧ (f dτ + dy)
K
1
Φ = Φ0 − ln K .
2
B =
• The final step is to take the near-horizon limit in order to compare with
the solutions of [26, 27]. So, h → R2/ρ2 . We switch variables to the
global radial coordinate
r
R
= ,
R
ρ
where the boundary now lies at r = 0 and the horizon at rH =
Using the parameter ∆ = βR2 we have
β 2 ρ2 =
∆2
,
r2
h=
r2
,
R2
f =1−
r4
4 ,
rH
K =1+
R2/RH .
∆2 r 2
4
rH
94
Null Melvin Twist
and the metric becomes
2
ds
=
0
3
R2
∆2
2
−dτ
1
+
r2 K
r2
+Kd,x + K
4
f + dy
2
3
∆2
1− 2 f
r
4
∆2
+ 2dτ dy 2 f
r
1
.
/
dr2
+ r2 Kds2P2 + (dχ + A)2
f
R2
(dχ + A) ∧ (f dτ + dy)
r2 K
1
Φ =Φ 0 − ln K .
2
B = 2∆
K varies smoothly between the boundary and the horizon between 1 and
2 .
1 + ∆2/rH
An important fact is that as f → 0 and Bt → 0, the surface
r = rH remains a non-singular null horizon. Null geodesics which span the
horizon have a perpendicular timelike Killing vector ∂τ near the horizon. This
means that we have a non-rotating black hole. This is interesting because
the geometry is not static but only stationary∗ and therefore we might have
expected a Killing horizon outside a black hole.
We call stationary, any metric that possesses a Killing vector that is timelike near
infinity. A metric is called static if it possesses a timelike Killing vector that is orthogonal
to a family of hypersurfaces. Physically, by stationary we mean that something is “doing
the same thing at every time” while static we mean “doing nothing at all”.
∗
D
Kaluza-Klein
Compactifications
The Kaluza-Klein theory was first formulated as an attempt to unify gravity and electromagnetism. In 1921 Kaluza[43] proposed that gravitation and
electromagnetism could be unified in a theory of five-dimensional Riemannian
geometry. Later, in 1926, Oscar Klein [44] suggested that the fourth spatial
dimension is curled up in a circle of small radius and took the original idea further. In this settting, a particle moving a short distance along that dimension
would return to its initial position. When we have a spacetime with such compact dimensions we talk about compactification. In the 1970’s this approach
was revived by Scherk and Schwarz and by Cremmer and Scherk, and as extra
true dimensions became necessary for a variety of theories, Kaluza-Klein compactification evolved into a commonly used tool for dimensional reduction. A
classical review on the subject, from a modern point o view, and in context
with supergravity is by Duff, Nillson and Pope[45].
Let us start describing the mechanism directly in D-dimensional spacetime
(D = d + 1) . We consider the case with xd being periodic, i.e.
xd = xd + 2πR ,
and with the remaining dimensions, xµ for µ = 0, . . . , d noncompact. This is
known as toroidal compactification. The d-dimensional metric then separates
into gµν , gµd and gdd which are effectively, the four-dimensional metric, a
vector (the gauge field) and a scalar.
Then the metric ansatz∗ is
'
D
M
N
µ
ν
d
µ
ds2 = gM
N dx dx = gµν dx dx + gdd dx + Aµ dx
or in block form
gM N (x , gdd ) =
µ
3
gµν
Aµ
Aµ
gdd
4
(2
,
.
The fields gµν , gdd and Aµ can only depend on the noncompact coordinates
and in d-dimensional actions indexes are raised and lowered with gµν only. The
action then becomes
ˆ
√
SD = dxD −gR(g(D) ) .
∗
where, in an obvious notation, capital Latin letters refer to D dimensions and Greek
indexes run over noncompact dimensions 0, . . . , d − 1.
95
96
Kaluza-Klein Compactifications
Then, noticing that the Ricci scalar “splits” into a d dimensional part Rd and
a part with Fµν , then by varying the action with respect to Aµ one obtains
Maxwell’s equations (dF = 0 and dF = 0) and by varying with respect to the
d-dimensional metric recovers Einstein’s equations
1
1
Rµν − Rgµν = 2 Tµν ,
2
Λ
where Tµν is the energy-momentum tensor and is equal to Tµν = F µν F κσ gνσ −
1/4g µκ |F |2 .
The metric ansatz is the most general metric invariant under translations
of xd . This allows d-dimensional reparameterizations x%µ (xν ) and
x%d = xd + λ(xµ ) .
This leads to
A%µ = Aµ − ∂µ λ .
So, gauge transformations arise as part of the higher dimensional coordinate
invariance. This is the so-called Kaluza-Klein mechanism.
Let us consider a massless scalar φ in D dimensions with gdd = 1 for
simplicity. Then, the momentum in the periodic dimension is quantized, pd =
n/R. We can expand the xd dependence of φ in a complete set
∞
2
3
inxd
φ(x ) =
φn (x ) exp
R
n=−∞
M
µ
4
.
The D-dimensional wave equation becomes
∂µ ∂ µ φn (xµ ) =
n2
φn (xµ ) .
R2
The modes of the D-dimensional field become an infinite tower of d dimensional fields labeled by n. The d-dimensional mass-squared is non-zero for all
fields with non-vanishing pd
−pµ pµ =
n2
.
R2
Let us now discuss what we have seen so far in more modern terms. To
make things concrete, we now restrict ourselves in five dimensions.´We effec√
tively start with the five-dimensional Einstein-Hilbert action S = d5 x gR
and instead of assuming that the ground state of this system is five-dimensional
Minkowski space (M 5 ), we take the ground state to be the product of fourdimensional Minkowski M 4 space and the unit circle S 1 , M 4 × S 1 . We choose
to do this, although it is difficult to classically to decide which of the two
spaces is appropriate. One assumes that the radius of the circle is microscopically small (e.g. the order of Planck scale) and this accounts for the fact the
existence of the extra dimension is not observed.
Kaluza-Klein Compactifications
97
In this picture the physical spectrum is determined by studying small
oscillations around this ground state.
When we examine the symmetries of the ground state M 4 × S 1 , we find
that we have four-dimensional Poincaré symmetry acting on M 4 and a U(1)
symmetry on S 1 . These symmetries appear as gauge symmetries in the fourdimensional space. The massless modes that emerge turn out to be a spin-two
graviton and a spin-one photon.
So, in principle, if we choose an appropriate higher-dimensional manifold
X we can exploit its symmetries to construct an effective four-dimensional
world with the desired gauge symmetries. So, the generalization of the ansatz
for a higher-dimensional compact space would correspond to an ansatz of the
form
gM N (xα , φk ) =
3
gµν (xα )
<
a α
a k
a Aµ (x )Ki (φ )
<
a α
a k
a Aµ (x )Ki (φ )
k
γij (φ )
4
.
This is for an n-dimensional compact manifold X where φi , i = 1, . . . , n are
the coordinates X. We have assumed generators of the symmetry group of
X, T a , a = 1, . . . , N . The symmetry generator on the φi acts as φi → φi +
Kia (φ), where Kia (φ) is a Killing vector associated with the symmetry T a . By
Killing vector we mean that the ground-state metric on X has a vanishing Lie
derivative with respect to Kia , £K γij = 0. So, Aaµ (xα ) are the massless gauge
fields of the symmetry group of X and γij is the metric tensor of the space X.
In this way, it is possible to obtain arbitrary Abelian or non-Abelian gauge
group as components of a gravitational field in 4 + n dimensions.
Although a realistic unified theory does not arise by this considerations
alone (nor by any other known to date) there are some interesting conclusion
that can be drawn, just by symmetry considerations on the compact manifold.
The gauge group we would like to obtain is obviously SU(3) × SU(2) × U(1).
So the symmetry group G of the compact space X must contain the Standard
Model group at least as a subgroup. In [46] we find a clear exposition of the
reasoning that follows.
For any symmetry group G the space of lowest dimension is a homogeneous
space G/H, where H is a maximal subgroup of G. In the case of G = SU(3) ×
SU(2) × U(1) the subgroup with the largest possible dimension that is suitable
is SU(2) × U(1) × U(1). Any larger subgroup would no longer be a symmetry
of the group G/H. The dimensionality that emerges for SU(3) × SU(2) ×
U(1)/SU(2) × U(1) × U(1) is therefore (8 + 3 + 1) − (3 + 1 + 1) = 7† . This is a
remarkable result, since it suggests that the dimensionality of the space M 4 ×
X is necessarily at least eleven. This agrees with the result in supergravity and
string theory. If we were to consider higher than eleven dimensions we would
have to include a massless particle of spin higher than two. But there are
†
Since the dimensionality of G/H is determined by the dimension of G minus the dimension of H.
98
Kaluza-Klein Compactifications
several reasons why higher than two spin massless particles coupled to gravity
do not exist. So, eleven dimensions are not only just enough to include the
Standard Model, but the also upper threshold if we take into consideration
field theory reasoning.
As a more concrete example of this we can consider the following. We
already saw that U(1) symmetry is already obtained if we consider the circle
S 1 . The lowest dimensional space with SU(2) symmetry is the ordinary twodimensional sphere S 2 . For the SU(3) symmetry the lowest dimensional space
is the complex projective space CP2 (Appendix C) which has real dimension
four. So, the space CP2 × S 2 × S 1 has the desired SU(3) × SU(2) × U(1)
symmetry and has 4 + 2 + 1 = 7 dimensions. Although it is shown in [46] that
the proper group is possible to achieve, there are still aspects of the Standard
Model that are unattainable by KK compactification. Also, there are several
issues that emerge (e.g. the instability of the KK ground state [47] and the
fact that it is impossible to obtain chiral fermions) if the Kaluza-Klein theory
is approached as an isolated framework of obtaining a unified four-dimensional
theory. As a tool however, it remains rather popular for compactifications in
the context of string theory or supergravity.
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